Understanding Slope

Slope is a measure of how steep a line is and the direction in which the line moves as you travel from left to right.

Slope describes the rate of change between two quantities. It tells us how much the vertical value changes for each unit of horizontal change.

For example:

• a steep hill has a large slope
• a flat road has a slope of zero
• a line rising from left to right has a positive slope
• a line falling from left to right has a negative slope

Slope is one of the most important ideas for understanding linear relationships.

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Slope as Steepness

Imagine several ramps.

A ramp that rises quickly over a short horizontal distance is:

steep

A ramp that rises slowly over a long horizontal distance is:

less steep

Slope gives us a mathematical way to compare this steepness.

A greater absolute value of slope means a steeper line.

For example:

m = 5 is steeper than m = 2

m = −6 is steeper than m = −3

The negative sign describes direction, not how steep the line is.

Therefore, when comparing steepness, consider:

|m|

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Slope as Direction

Slope also tells us which way a line moves as x increases.

Reading a graph from left to right:

line rises → positive slope

line falls → negative slope

line remains horizontal → zero slope

line is vertical → undefined slope

These four slope types are fundamental.

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Rise Over Run

Slope is often described as:

slope = rise/run

The rise is the vertical change.

The run is the horizontal change.

Therefore:

m = change in y/change in x

or:

m = Δy/Δx

The symbol Δ means:

change in

So:

Δy = change in y

Δx = change in x

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Calculating Slope from Two Points

If two points on a line are:

(x₁, y₁) and (x₂, y₂)

the slope is:

m = (y₂ − y₁)/(x₂ − x₁)

The numerator calculates the vertical change.

The denominator calculates the horizontal change.

An important rule is:

subtract the coordinates in the same order.

If you calculate y₂ − y₁ in the numerator, you must calculate x₂ − x₁ in the denominator.

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Worked Example: Positive Slope

Find the slope between:

(2, 3) and (6, 11)

Use:

m = (y₂ − y₁)/(x₂ − x₁)

Substitute:

m = (11 − 3)/(6 − 2)

m = 8/4

m = 2

The slope is positive.

This means that for every increase of 1 in x:

y increases by 2

The line rises from left to right.

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Understanding the Answer

A slope of:

m = 2

can also be written:

m = 2/1

This means:

rise = 2

run = 1

So you can move along the line by going:

right 1, up 2

Other equivalent movements include:

right 2, up 4

right 3, up 6

right 5, up 10

They all have the same ratio:

rise/run = 2

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Worked Example: Negative Slope

Find the slope between:

(1, 8) and (5, 2)

m = (2 − 8)/(5 − 1)

m = −6/4

m = −3/2

The negative slope means the line falls from left to right.

For every 2 units moved to the right:

y decreases by 3

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Understanding Negative Slope

A slope of:

m = −3/2

can be interpreted as:

right 2 → down 3

because:

rise = −3

run = 2

You could also write:

3/−2

which represents:

left 2 → up 3

Both movements follow the same line.

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The Four Types of Slope

Every straight line has one of four slope classifications.

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Positive Slope

A line has a positive slope when y increases as x increases.

For example:

m = 3

means:

y increases by 3 for every increase of 1 in x

Positive slopes rise from:

lower left → upper right

Examples of positive rates might include:

• money earned as hours worked increase
• distance travelled as time increases
• water level rising as water is added

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Negative Slope

A line has a negative slope when y decreases as x increases.

For example:

m = −4

means:

y decreases by 4 for every increase of 1 in x

Negative slopes fall from:

upper left → lower right

Examples might include:

• fuel remaining as distance travelled increases
• money remaining as purchases are made
• altitude decreasing during a steady descent

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Zero Slope

A horizontal line has:

m = 0

Suppose two points are:

(2, 5) and (8, 5)

Then:

m = (5 − 5)/(8 − 2)

m = 0/6

m = 0

There is no vertical change.

The y-value remains constant.

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Interpreting Zero Slope

A zero slope means:

no change in y as x changes

For example, imagine a tank containing exactly 500 L of water over a period when no water enters or leaves.

If:

x = time

y = water volume

then the graph would be horizontal.

Its slope would be:

0 L/min

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Undefined Slope

A vertical line has an:

undefined slope

Consider:

(4, 2) and (4, 9)

Then:

m = (9 − 2)/(4 − 4)

m = 7/0

Division by zero is undefined.

Therefore:

the slope is undefined

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Why Vertical Lines Have Undefined Slope

For a vertical line:

Δx = 0

The horizontal change is zero.

Since:

m = Δy/Δx

we would need to divide by zero.

Division by zero is not defined.

Therefore, a vertical line does not have a slope of zero.

It has:

undefined slope

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Zero vs Undefined Slope

These are often confused.

Horizontal line

Δy = 0

m = 0/Δx

m = 0

Vertical line

Δx = 0

m = Δy/0

m is undefined

A useful reminder is:

horizontal = zero

vertical = undefined

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Calculating Slope in Either Direction

Suppose the points are:

(2, 3) and (6, 11)

We previously calculated:

m = (11 − 3)/(6 − 2)

m = 8/4

m = 2

Now reverse the order:

m = (3 − 11)/(2 − 6)

m = −8/−4

m = 2

The answer is unchanged.

You may subtract the points in either order as long as you use:

the same order in the numerator and denominator.

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A Common Calculation Error

Suppose the points are:

(2, 3) and (6, 11)

Incorrect:

m = (11 − 3)/(2 − 6)

m = 8/−4

m = −2

The mistake is that the subtraction order changed between the numerator and denominator.

Correct:

m = (11 − 3)/(6 − 2)

or:

m = (3 − 11)/(2 − 6)

Both give:

m = 2

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Slope from a Graph

You can also calculate slope directly from a graph.

Choose two clear points on the line.

Then determine:

rise = vertical change

run = horizontal change

Finally:

m = rise/run

For greater accuracy, choose points that lie exactly on grid intersections when possible.

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The Slope Triangle

A slope triangle can help visualize rise and run.

Starting from one point:

1. move horizontally to the x-coordinate of the second point
2. move vertically to reach the second point

The horizontal side represents:

run

The vertical side represents:

rise

Then:

m = rise/run

Different slope triangles along the same straight line produce equivalent ratios.

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Example from a Graph

Suppose moving from one point to another requires:

right 4

up 6

Then:

rise = 6

run = 4

m = 6/4

m = 3/2

If another larger triangle gives:

rise = 12

run = 8

then:

m = 12/8

m = 3/2

The slope remains constant because the line is straight.

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Slope as a Rate of Change

Slope does more than describe the appearance of a graph.

It represents:

rate of change

In general:

slope = change in output/change in input

If x represents time and y represents distance:

slope = change in distance/change in time

This is a speed or rate of travel.

If x represents hours worked and y represents money earned:

slope = change in money/change in hours

This represents earnings per hour.

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Units of Slope

Slope often has meaningful units.

Suppose:

x = time in hours

y = distance in kilometres

Then:

slope = kilometres/hours

The units are:

km/h

Suppose:

x = number of tickets

y = total cost in dollars

Then:

slope = dollars/ticket

The units are:

$/ticket

Always interpret slope using:

y-units per x-unit

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Worked Example: Distance and Time

A cyclist has travelled:

20 km after 1 hour

and:

60 km after 3 hours

The points are:

(1, 20) and (3, 60)

Calculate:

m = (60 − 20)/(3 − 1)

m = 40/2

m = 20

Units:

km/h

Interpretation:

The cyclist's distance increases by:

20 km for every additional hour

over this interval.

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Worked Example: Earning Money

A worker earns:

$45 after 3 hours

and:

$105 after 7 hours

Slope:

m = (105 − 45)/(7 − 3)

m = 60/4

m = 15

Units:

$/hour

Interpretation:

The worker earns:

$15 for each additional hour worked

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Worked Example: Decreasing Quantity

A tank contains:

800 L at 2 minutes

and:

500 L at 5 minutes

The points are:

(2, 800) and (5, 500)

Calculate:

m = (500 − 800)/(5 − 2)

m = −300/3

m = −100 L/min

The negative sign is important.

Interpretation:

The amount of water decreases by:

100 L each minute

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Positive Slope in Context

A positive slope means the dependent variable increases as the independent variable increases.

Examples:

time increases → distance increases

hours worked increase → earnings increase

items sold increase → revenue increases

water added increases → volume increases

A positive slope therefore represents an:

increasing relationship

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Negative Slope in Context

A negative slope means the dependent variable decreases as the independent variable increases.

Examples:

time increases → battery charge decreases

distance driven increases → fuel remaining decreases

time increases → amount of medicine in the body may decrease during a simplified interval

A negative slope represents a:

decreasing relationship

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Zero Slope in Context

Zero slope means the output remains constant as the input changes.

Examples:

time passes → temperature remains constant

time passes → parked object's position remains constant

number of observations increases → a fixed reference value remains unchanged

The graph is horizontal.

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Comparing Positive Slopes

Suppose two lines have slopes:

Line A: m = 2

Line B: m = 5

Both are positive.

Both rise from left to right.

However:

Line B is steeper

because:

|5| > |2|

It rises more rapidly for the same horizontal change.

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Comparing Negative Slopes

Suppose:

Line A: m = −2

Line B: m = −7

Which is steeper?

Compare absolute values:

|−2| = 2

|−7| = 7

Therefore:

Line B is steeper

Although:

−7 < −2

the line with slope −7 has the greater steepness.

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Comparing Positive and Negative Slopes

Suppose:

Line A: m = 4

Line B: m = −4

They have:

equal steepness

because:

|4| = |−4|

But they have opposite directions.

Line A rises from left to right.

Line B falls from left to right.

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Comparing Lines Visually

If two lines are drawn on the same coordinate scale, their steepness can be compared visually.

A line closer to vertical generally has a greater absolute slope.

A line closer to horizontal has a smaller absolute slope.

However, visual comparison can be misleading if separate graphs use different axis scales.

The safest method is:

calculate and compare the slopes.

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Parallel Lines

Distinct nonvertical parallel lines have:

the same slope

For example:

y = 3x + 2

and:

y = 3x − 5

both have:

m = 3

They never meet because they rise at exactly the same rate while having different vertical positions.

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Slope and Linear Relationships

A straight line has a:

constant slope

This means its rate of change remains constant.

If a line has:

m = 4

then every increase of 1 in x produces an increase of 4 in y.

This constant rate of change is one of the defining features of a linear relationship.

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Slope in y = mx + b

A common form of a linear equation is:

y = mx + b

where:

m = slope

b = y-intercept

For example:

y = 3x + 5

has:

m = 3

Therefore, y increases by 3 whenever x increases by 1.

The value 5 tells us where the line crosses the y-axis.

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Example: Reading Slope from an Equation

Consider:

y = −2x + 7

The coefficient of x is:

−2

Therefore:

m = −2

The line has a negative slope.

It falls from left to right.

For every increase of 1 in x:

y decreases by 2

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Fractional Slopes

Slope does not need to be a whole number.

For example:

m = 2/3

means:

right 3 → up 2

A slope of:

m = −1/4

means:

right 4 → down 1

Fractional slopes often describe lines that are less steep than lines with large whole-number slopes.

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Decimal Slopes

Slope can also be expressed as a decimal.

For example:

m = 0.5

is equivalent to:

m = 1/2

This means:

right 2 → up 1

Similarly:

m = −0.25

is equivalent to:

m = −1/4

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Slope and Constant Rate

Consider this table:

Each time x increases by 1:

y increases by 3.

Therefore:

slope = 3

Because the rate of change is constant, the relationship is linear.

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Unequal x-Intervals

Be careful when x-values do not increase by equal amounts.

Consider:

From the first to second point:

Δy = 6

Δx = 2

m = 6/2 = 3

From the second to third:

Δy = 12

Δx = 4

m = 12/4 = 3

The y-values do not increase by the same amount, but the rate of change is still constant.

Therefore, the relationship is linear.

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Slope and Average Rate of Change

For a straight line, slope is constant everywhere.

For a curved graph, however, the slope between two points represents the:

average rate of change over that interval

For example, if a car's distance-time graph curves, the car's speed is changing.

Calculating the slope between two points gives the average rate over that interval rather than a single constant rate for the entire journey.

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Real-World Example: Temperature

Suppose the temperature is:

18°C at 8:00 a.m.

and:

24°C at 11:00 a.m.

Let x represent hours after 8:00 a.m.

The points are:

(0, 18) and (3, 24)

m = (24 − 18)/(3 − 0)

m = 6/3

m = 2°C/hour

Interpretation:

The temperature increased at an average rate of:

2°C per hour

during that period.

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Real-World Example: Fuel

A vehicle has:

50 L of fuel at 0 km

and:

38 L after 180 km

Slope:

m = (38 − 50)/(180 − 0)

m = −12/180

m ≈ −0.067 L/km

The negative slope means fuel is being used.

A more convenient interpretation might be:

approximately 6.7 L per 100 km consumed over this interval.

The context determines the most useful way to express a rate.

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Slope and Units Matter

Suppose a student calculates:

m = 12

This answer is incomplete in many real-world problems.

What does 12 mean?

It could mean:

• $12/hour
• 12 km/h
• 12 L/min
• 12°C/hour
• 12 people/year

A meaningful interpretation should include both:

value and units

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Slope Does Not Always Mean Speed

Slope means:

rate of change

Speed is only one possible rate.

Depending on the graph, slope could represent:

• speed
• acceleration
• cost per item
• hourly wage
• population growth
• cooling rate
• water flow
• fuel consumption
• change in height
• change in temperature

Always examine the axis labels.

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Choosing Two Points

When calculating slope from a graph, choose points carefully.

Good points:

• lie exactly on the line
• are easy to read
• preferably lie on grid intersections
• are reasonably far apart

Points farther apart can reduce the effect of small reading errors on a graph.

Do not automatically use points that are not actually on the line.

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A Complete Comparison Example

Consider four lines:

Line A: m = 4

Line B: m = 1

Line C: m = −3

Line D: m = 0

Line A rises more steeply than Line B.

Line C falls from left to right.

Its steepness is greater than Line B because:

|−3| > |1|

Line D is horizontal.

Its slope is zero.

If a fifth line were vertical, its slope would be undefined.

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A Reliable Strategy for Finding Slope

When given two points:

1. Label the points (x₁, y₁) and (x₂, y₂).
2. Write the slope formula.
3. Calculate y₂ − y₁.
4. Calculate x₂ − x₁.
5. Keep the subtraction order consistent.
6. Divide.
7. Simplify the fraction if possible.
8. Determine whether the slope is positive, negative, zero, or undefined.
9. Include units if the problem has a real-world context.
10. Interpret what the slope means.

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A Reliable Strategy for Interpreting Slope

When given a slope in context, ask:

1. What does x represent?
2. What does y represent?
3. What are the x-units?
4. What are the y-units?
5. Is the slope positive or negative?
6. Is the relationship increasing or decreasing?
7. What does one unit of x correspond to?
8. What does the slope tell us about the change in y?
9. Are the data continuous or discrete?
10. Is the slope constant or only an average over an interval?

A good interpretation often follows:

"For every 1 [x-unit], y increases/decreases by [slope magnitude] [y-units]."

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Common Mistakes

Reversing only one subtraction

If you reverse the y-values, you must also reverse the x-values.

Using x-change over y-change

Slope is:

change in y/change in x

not the reverse.

Ignoring negative signs

A negative slope communicates important information about direction.

Thinking −6 is less steep than −2 because −6 is a smaller number

Steepness depends on absolute value.

|−6| > |−2|

so −6 is steeper.

Saying a vertical line has slope zero

Horizontal lines have zero slope. Vertical lines have undefined slope.

Dividing by zero

If Δx = 0, the slope is undefined.

Forgetting units

Real-world slopes should usually be interpreted with y-units per x-unit.

Assuming every slope represents speed

Slope represents rate of change. Its meaning depends on the variables.

Comparing visual steepness on graphs with different scales

Different axis scales can make identical slopes look different.

Assuming equal changes in y prove a relationship is linear

You must consider the corresponding changes in x. The ratio Δy/Δx must remain constant.

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Key Terms

Slope — measure of the steepness and direction of a line.

Rate of change — amount one variable changes compared with the change in another variable.

Rise — vertical change between two points.

Run — horizontal change between two points.

Δy — change in the y-coordinate.

Δx — change in the x-coordinate.

Positive slope — slope greater than zero; the line rises from left to right.

Negative slope — slope less than zero; the line falls from left to right.

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line, caused by division by zero.

Steepness — how rapidly a line rises or falls relative to horizontal change.

Absolute value — distance from zero; useful when comparing slope steepness.

Linear relationship — relationship with a constant rate of change and a straight-line graph.

Constant rate of change — rate that remains the same throughout a linear relationship.

Slope triangle — right triangle used on a graph to show rise and run.

y-intercept — point where a line crosses the y-axis.

Average rate of change — change in output divided by change in input over an interval.

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Key Takeaways

• Slope measures both the steepness and direction of a line.
• Slope can be described as rise over run.
• Slope is calculated using m = Δy/Δx.
• From two points, use m = (y₂ − y₁)/(x₂ − x₁).
• Coordinate subtraction must be performed in the same order.
• A positive slope rises from left to right.
• A negative slope falls from left to right.
• A horizontal line has zero slope.
• A vertical line has undefined slope.
• A larger absolute slope means a steeper line.
• The sign of slope describes direction.
• The absolute value of slope helps describe steepness.
• Straight lines have constant slope.
• Distinct nonvertical parallel lines have equal slopes.
• Slope represents a rate of change between two variables.
• In real-world contexts, slope should be interpreted using units.
• Slope units are y-units per x-unit.
• A positive slope represents an increasing relationship.
• A negative slope represents a decreasing relationship.
• Zero slope represents a constant output.
• Different-looking graphs should not be compared visually unless their scales are the same.
• Slope does not always represent speed; its meaning depends on the variables.
• For nonlinear graphs, the slope between two points represents an average rate of change over that interval.
• Understanding slope provides the foundation for graphing linear equations and interpreting mathematical models.

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Check Your Understanding

1. Explain what slope tells us about a line.
2. Describe the difference between steepness and direction when interpreting slope.
3. Calculate the slope between the points (2, 5) and (6, 13).
4. Calculate the slope between (1, 9) and (5, 1). Classify the slope.
5. Find the slope between (−3, 4) and (5, 4). Explain what the result tells you about the line.
6. Find the slope between (7, 2) and (7, 10). Explain why the slope is undefined.
7. Which line is steeper: m = 3 or m = 7? Explain.
8. Which line is steeper: m = −8 or m = −4? Explain why simply comparing −8 and −4 numerically can be misleading.
9. Compare lines with slopes m = 5 and m = −5. How are their steepness and direction related?
10. A cyclist travels 24 km after 1 hour and 72 km after 3 hours. Calculate the slope and interpret its meaning.
11. A water tank contains 900 L after 2 minutes and 600 L after 5 minutes. Calculate the slope, include units, and explain the meaning of the negative sign.
12. A student's earnings are represented by y = 18x + 25, where x is hours worked and y is dollars. Identify the slope and explain its real-world meaning.
13. Two distinct lines both have slope −3. What can you conclude about the relationship between the lines?
14. A graph appears steeper than another graph, but the two graphs use different axis scales. Explain why you should not rely only on their appearance to compare slopes.
15. A student calculates the slope between (2, 4) and (8, 16) as (16 − 4)/(2 − 8) = −2. Identify the student's mistake and calculate the correct slope.

Slope-Intercept Form

Slope-intercept form is one of the most useful ways to write the equation of a straight line because it immediately tells us two important features of the graph:

y = mx + b

where:

• m = slope
• b = y-intercept

The slope tells us the line's steepness and direction, while the y-intercept tells us where the line crosses the y-axis.

For example:

y = 3x + 2

has:

slope = 3

y-intercept = 2, or the point (0, 2)

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Understanding y = mx + b

Consider:

y = 2x + 5

Compare this with:

y = mx + b

We can identify:

m = 2

b = 5

Therefore:

• the slope is 2
• the line rises from left to right
• y increases by 2 whenever x increases by 1
• the line crosses the y-axis at (0, 5)

Slope-intercept form gives us enough information to graph the line without making a complete table of values.

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The Slope

In:

y = mx + b

the coefficient of x is the slope.

For example:

y = 4x + 7

has:

m = 4

Since:

4 = 4/1

we can interpret the slope as:

right 1, up 4

The positive slope means the line rises from left to right.

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The y-Intercept

The y-intercept is the point where the graph crosses the y-axis.

Every point on the y-axis has:

x = 0

If:

y = mx + b

substitute x = 0:

y = m(0) + b

Therefore:

y = b

So the y-intercept is:

(0, b)

This explains why b represents the y-intercept.

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Identifying Slope and y-Intercept

Consider:

y = 5x − 3

The coefficient of x is:

5

Therefore:

m = 5

The constant is:

−3

Therefore:

b = −3

The y-intercept is:

(0, −3)

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Be Careful with Negative Signs

Consider:

y = −4x + 6

The slope is:

m = −4

The y-intercept is:

b = 6

Now consider:

y = 4x − 6

The slope is:

m = 4

The y-intercept is:

b = −6

The negative sign can belong to either m or b, so read the equation carefully.

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Equations with No Written Constant

Consider:

y = 3x

This can be written as:

y = 3x + 0

Therefore:

m = 3

b = 0

The y-intercept is:

(0, 0)

The line passes through the origin.

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Equations with No x-Term

Consider:

y = 7

This can be thought of as:

y = 0x + 7

Therefore:

m = 0

b = 7

The graph is a horizontal line crossing the y-axis at:

(0, 7)

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Fractional Slopes

Suppose:

y = (2/3)x + 1

Then:

m = 2/3

b = 1

The slope tells us:

rise = 2

run = 3

Starting from the y-intercept, we can move:

right 3, up 2

to locate another point.

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Negative Fractional Slopes

Consider:

y = −(3/4)x + 5

Then:

m = −3/4

b = 5

Starting at:

(0, 5)

move:

right 4, down 3

This gives another point on the line.

The negative slope means the line falls from left to right.

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Graphing from Slope-Intercept Form

To graph:

y = mx + b

use this method:

1. Identify m.
2. Identify b.
3. Plot the y-intercept (0, b).
4. Write the slope as a fraction.
5. Use the numerator as the rise.
6. Use the denominator as the run.
7. Plot another point.
8. Repeat the slope movement if useful.
9. Draw a straight line through the points.
10. Extend the line in both directions.

This method is usually much faster than creating a large table of values.

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Worked Example: Graphing a Positive Slope

Graph:

y = 2x + 1

Identify:

m = 2

b = 1

First plot the y-intercept:

(0, 1)

Write the slope as:

2/1

From (0, 1):

right 1

up 2

This gives:

(1, 3)

Repeat:

right 1

up 2

This gives:

(2, 5)

Draw a straight line through the points.

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Checking with Substitution

We can verify that our points satisfy:

y = 2x + 1

For (1, 3):

3 = 2(1) + 1

3 = 3

Correct.

For (2, 5):

5 = 2(2) + 1

5 = 5

Correct.

Substitution is a useful way to check points on a graph.

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Worked Example: Graphing a Negative Slope

Graph:

y = −3x + 4

Identify:

m = −3

b = 4

Plot:

(0, 4)

Write:

m = −3/1

From (0, 4):

right 1

down 3

This gives:

(1, 1)

Repeat:

right 1

down 3

This gives:

(2, −2)

The resulting line falls from left to right.

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Graphing a Fractional Slope

Graph:

y = (1/2)x − 3

Identify:

m = 1/2

b = −3

Start at:

(0, −3)

Use the slope:

right 2

up 1

This gives:

(2, −2)

Repeat:

right 2

up 1

This gives:

(4, −1)

Then draw the line.

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Moving Backward Along the Line

Slope can also be used in reverse.

Suppose:

m = 2/3

Normally we might move:

right 3

up 2

But we can also move:

left 3

down 2

because:

−2/−3 = 2/3

Both movements stay on the same line.

This is useful when the graph has limited space.

---

Writing an Equation from a Graph

A graph can also be converted into slope-intercept form.

To write the equation:

1. Find the y-intercept.
2. Determine the slope.

3. Substitute both values into:

   y = mx + b

For example, suppose a graph crosses the y-axis at:

(0, 3)

and rises 2 units for every 1 unit moved right.

Then:

m = 2

b = 3

Therefore:

y = 2x + 3

---

Finding Slope from Two Points on a Graph

Sometimes the slope is not immediately obvious.

Choose two points on the line.

Suppose:

(0, 2) and (4, 10)

Calculate:

m = (10 − 2)/(4 − 0)

m = 8/4

m = 2

The y-intercept is already visible:

b = 2

Therefore:

y = 2x + 2

---

Worked Example: Equation from a Graph

Suppose a line crosses the y-axis at:

(0, 6)

and also passes through:

(2, 2)

Calculate slope:

m = (2 − 6)/(2 − 0)

m = −4/2

m = −2

The y-intercept is:

b = 6

Therefore:

y = −2x + 6

---

When the y-Intercept Is Negative

Suppose a graph crosses the y-axis at:

(0, −4)

and has slope:

3

Then:

m = 3

b = −4

Therefore:

y = 3x − 4

Do not write:

y = 3x + 4

The sign of the y-intercept must match the graph.

---

Finding an Equation from Two Points

If one point is the y-intercept, the process is especially easy.

Suppose a line passes through:

(0, −2) and (3, 4)

Calculate:

m = (4 − (−2))/(3 − 0)

m = 6/3

m = 2

Since:

(0, −2)

is the y-intercept:

b = −2

Therefore:

y = 2x − 2

---

What if the y-Intercept Is Not Given?

Suppose a line passes through:

(2, 5) and (4, 9)

First find the slope:

m = (9 − 5)/(4 − 2)

m = 4/2

m = 2

Now use:

y = mx + b

Substitute one known point:

5 = 2(2) + b

5 = 4 + b

b = 1

Therefore:

y = 2x + 1

This equation can then be checked using the other point.

---

Changing the Slope

Consider these equations:

y = x + 2

y = 2x + 2

y = 4x + 2

All have:

b = 2

Therefore, all three lines cross the y-axis at:

(0, 2)

But their slopes are different.

As the positive slope increases:

the line becomes steeper

---

Positive Slope Changes

Compare:

m = 1/2

m = 1

m = 3

All are positive.

All lines rise from left to right.

However:

m = 3

is the steepest.

The larger the absolute value of the slope:

the steeper the line

---

Negative Slope Changes

Compare:

m = −1

m = −3

m = −6

All lines fall from left to right.

The line with:

m = −6

is the steepest because:

|−6| = 6

The negative sign tells us direction.

The absolute value tells us steepness.

---

Changing the Sign of the Slope

Compare:

y = 3x + 2

and:

y = −3x + 2

Both have the same y-intercept:

(0, 2)

Both have the same steepness because:

|3| = |−3|

But they move in opposite directions.

The first rises.

The second falls.

---

When the Slope Is Zero

Consider:

y = 0x + 4

This simplifies to:

y = 4

The graph is horizontal.

Therefore:

m = 0

A zero slope means y does not change as x changes.

---

Can a Vertical Line Be Written in Slope-Intercept Form?

A vertical line has an undefined slope.

For example:

x = 3

is a vertical line.

It cannot be written in the form:

y = mx + b

because there is no defined value of m.

Therefore, slope-intercept form describes nonvertical straight lines, including horizontal lines, but not vertical lines.

---

Changing the y-Intercept

Now consider:

y = 2x + 1

y = 2x + 4

y = 2x − 3

All three lines have:

m = 2

Therefore, they have the same steepness and direction.

But they cross the y-axis at different locations.

---

What the y-Intercept Does to the Graph

Changing b moves the line:

up or down

without changing its slope.

Increasing b shifts the line upward.

Decreasing b shifts the line downward.

For example:

y = 2x + 5

lies above:

y = 2x + 1

for every x-value.

The lines remain parallel because their slopes are equal.

---

Same Slope, Different y-Intercepts

Consider:

y = 3x + 5

and:

y = 3x − 2

Both have:

m = 3

Therefore, they are distinct parallel lines.

Their y-intercepts are:

(0, 5)

and:

(0, −2)

Changing the y-intercept changes the vertical position of the line without changing its direction or steepness.

---

Same y-Intercept, Different Slopes

Consider:

y = x + 4

and:

y = −2x + 4

Both cross the y-axis at:

(0, 4)

But:

first line: m = 1

second line: m = −2

Therefore, the lines have different directions and steepness.

They intersect at their shared y-intercept.

---

Comparing m and b

The two parameters have different jobs.

---

Parallel Lines and Slope-Intercept Form

Consider:

y = 4x + 1

y = 4x − 5

Both have:

m = 4

but different y-intercepts.

Therefore, the lines are:

parallel

For distinct nonvertical lines:

same slope + different y-intercepts → parallel lines

---

Intersecting Lines

Consider:

y = 2x + 1

and:

y = 5x + 1

The slopes are different.

Therefore, the lines are not parallel.

They will intersect at exactly one point.

In this case they share the same y-intercept, so they intersect at:

(0, 1)

---

Slope-Intercept Form in Real-World Situations

Slope-intercept form is especially useful for situations involving:

constant rates of change

The general model:

y = mx + b

can be interpreted as:

output = rate × input + starting value

This is one of the most useful interpretations of slope-intercept form.

---

Real-World Example: Taxi Fare

Suppose a taxi charges:

• $4 starting fee
• $2 for every kilometre travelled

Let:

x = kilometres travelled

y = total cost

The equation is:

y = 2x + 4

Here:

m = 2

means:

$2 per kilometre

and:

b = 4

means:

$4 starting cost

---

Real-World Example: Savings

Suppose a person already has:

$100

and saves:

$25 each week

Let:

x = number of weeks

y = total savings

Then:

y = 25x + 100

The slope:

25

means savings increase by $25 each week.

The y-intercept:

100

means the person started with $100 at week 0.

---

Real-World Example: Water Tank

A tank initially contains:

500 L

Water enters at:

20 L/min

Let:

x = time in minutes

y = volume in litres

Then:

y = 20x + 500

Slope:

20 L/min

y-intercept:

500 L

The y-intercept represents the starting amount.

---

Real-World Example: Decreasing Quantity

A tank begins with:

900 L

and loses:

30 L/min

The equation is:

y = −30x + 900

The negative slope indicates that the amount decreases.

The y-intercept represents:

900 L at time 0

---

Interpreting the y-Intercept Carefully

Although b is always mathematically the value of y when x = 0, it may not always have a meaningful real-world interpretation.

Suppose x represents:

age of an adult

and the model is based only on ages 20–60.

The y-intercept would correspond to:

age 0

which is outside the model's intended domain.

Therefore, always ask:

Does x = 0 make sense in this situation?

---

Writing an Equation from a Real-World Description

Suppose a gym charges:

$30 membership fee

plus:

$8 per visit

Let:

x = number of visits

y = total cost

Rate:

m = 8

Starting amount:

b = 30

Therefore:

y = 8x + 30

Slope-intercept form makes the relationship between the rate and starting value very clear.

---

Writing an Equation from a Table

Consider:

The y-values increase by:

3

for every increase of 1 in x.

Therefore:

m = 3

When x = 0:

y = 5

Therefore:

b = 5

The equation is:

y = 3x + 5

---

A Table Without x = 0

Consider:

Calculate slope:

m = (15 − 9)/(4 − 2)

m = 6/2

m = 3

Use one point in:

y = mx + b

9 = 3(2) + b

9 = 6 + b

b = 3

Therefore:

y = 3x + 3

---

Checking an Equation Against a Graph

Suppose you think a graph represents:

y = −2x + 5

Check two features.

First:

Does the graph cross the y-axis at:

(0, 5)?

Second:

Does the graph move:

right 1, down 2?

If both are true, the equation is consistent with the graph.

---

Checking an Equation with a Point

Suppose the proposed equation is:

y = 3x − 2

and the graph contains:

(4, 10)

Substitute:

10 = 3(4) − 2

10 = 12 − 2

10 = 10

Therefore, the point lies on the line.

---

Graph Transformations Using m and b

The equation:

y = mx + b

can be thought of as a family of straight lines.

Changing m changes:

• steepness
• direction

Changing b changes:

• vertical position
• y-intercept

This allows us to predict graph changes without plotting many points.

---

Comparing Several Equations

Consider:

A: y = 2x + 3

B: y = 5x + 3

C: y = −2x + 3

D: y = 2x − 4

Compare A and B:

same y-intercept, different slopes

B is steeper.

Compare A and C:

same y-intercept, opposite slope signs

They rise and fall in opposite directions.

Compare A and D:

same slope, different y-intercepts

They are parallel.

---

From Graph to Equation: A Reliable Method

When writing an equation from a graph:

1. Locate the y-axis.
2. Find where the line crosses it.
3. Record that value as b.
4. Choose another clear point on the line.
5. Calculate rise and run.
6. Find m = rise/run.
7. Determine the sign of m.

8. Write:

   y = mx + b

9. Check the signs carefully.
10. Substitute a point to verify the equation.

---

From Equation to Graph: A Reliable Method

When graphing:

y = mx + b

1. Identify m.
2. Identify b.
3. Plot (0, b).
4. Write m as a fraction.
5. Use the numerator for vertical movement.
6. Use the denominator for horizontal movement.
7. Plot a second point.
8. Repeat if useful.
9. Draw the straight line.
10. Check whether its direction matches the sign of m.

If m > 0:

the graph should rise.

If m < 0:

the graph should fall.

If m = 0:

the graph should be horizontal.

---

Common Mistakes

Confusing m and b

In y = mx + b:

m = slope

b = y-intercept.

Giving only b instead of the y-intercept point

If b = 4, the y-intercept is:

(0, 4)

Plotting the y-intercept on the x-axis

The y-intercept must lie on the y-axis.

Ignoring the negative sign on the slope

m = −3 means the line falls from left to right.

Using slope backward

For m = 2/3:

rise = 2

run = 3

not rise = 3 and run = 2.

Thinking a negative y-intercept means negative slope

Slope and y-intercept are independent.

For example:

y = 4x − 7

has positive slope and negative y-intercept.

Thinking changing b changes steepness

Changing b moves the line vertically. It does not change slope.

Thinking changing m only moves the line up or down

Changing m changes the line's steepness and possibly its direction.

Thinking every straight line can be written in slope-intercept form

Vertical lines such as x = 4 have undefined slope and cannot be written as y = mx + b.

Assuming b always has a meaningful physical interpretation

In a model, x = 0 may lie outside the relevant domain.

---

Key Terms

Slope-intercept form — form of a linear equation written as y = mx + b.

Slope (m) — rate of change and measure of a line's steepness and direction.

y-intercept (b) — y-value where the line crosses the y-axis.

y-intercept point — coordinate (0, b).

Rise — vertical change between points.

Run — horizontal change between points.

Positive slope — slope greater than zero; graph rises from left to right.

Negative slope — slope less than zero; graph falls from left to right.

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line.

Linear equation — equation whose graph is a straight line.

Linear relationship — relationship with a constant rate of change.

Rate of change — amount one variable changes relative to another.

Parallel lines — lines in the same plane that never meet; distinct nonvertical parallel lines have equal slopes.

Origin — point (0, 0).

Coefficient — numerical factor multiplying a variable.

Constant — number without a variable attached.

Initial value — value of the dependent variable when the independent variable is zero, when meaningful in context.

---

Key Takeaways

• Slope-intercept form is y = mx + b.
• The value m represents slope.
• The value b represents the y-intercept.
• The y-intercept is the point (0, b).
• Positive slopes rise from left to right.
• Negative slopes fall from left to right.
• Larger absolute slope values produce steeper lines.
• A slope of zero produces a horizontal line.
• Vertical lines have undefined slope and cannot be written in slope-intercept form.
• To graph y = mx + b, plot the y-intercept first and then use the slope.
• A fractional slope can be interpreted using rise over run.
• A graph can be converted to an equation by finding its slope and y-intercept.
• If the y-intercept is not directly given, b can be calculated using a known point and slope.
• Changing m changes the line's steepness and direction.
• Changing b moves the line vertically without changing its slope.
• Distinct lines with the same slope and different y-intercepts are parallel.
• Lines with different slopes intersect at exactly one point.
• In real-world models, m often represents a rate.
• In real-world models, b often represents an initial or starting value.
• The y-intercept is not always meaningful if x = 0 lies outside the model's relevant domain.
• Slope-intercept form provides a direct connection between equations, graphs, tables, and real-world linear relationships.

---

Check Your Understanding

1. In the equation y = 4x + 7, identify the slope and y-intercept.
2. Identify m and b in y = −3x − 5. State whether the line rises or falls from left to right.
3. Find the slope and y-intercept of y = (2/5)x + 4.
4. Describe how you would graph y = 3x − 2 using only its slope and y-intercept.
5. Describe how you would graph y = −(2/3)x + 5.
6. A line crosses the y-axis at (0, 4) and has slope 3. Write its equation in slope-intercept form.
7. A line crosses the y-axis at (0, −6) and passes through (2, 0). Calculate its slope and write its equation.
8. A line passes through (2, 7) and (4, 13). Calculate the slope, determine the y-intercept, and write the equation in slope-intercept form.
9. Compare y = 2x + 1 and y = 6x + 1. What remains the same, and what changes?
10. Compare y = −3x + 5 and y = −3x − 4. Describe the relationship between their graphs.
11. Explain what happens to the graph of y = mx + b when b increases while m remains constant.
12. Explain what happens as the absolute value of m increases while b remains constant.
13. A taxi charges a $5 starting fee and $3 per kilometre. Write an equation for total cost y after travelling x kilometres. Explain the meaning of both m and b.
14. Explain why the vertical line x = 6 cannot be written in slope-intercept form.
15. A student says, "Changing the y-intercept makes a line steeper because the line moves higher on the graph." Evaluate this statement. Explain the different effects of changing m and changing b.

Point-Slope Form

Point-slope form is a useful way to write the equation of a straight line when you know:

• one point on the line
• the slope of the line

Instead of first finding the y-intercept, you can use the known point and slope directly.

The general form is:

y − y₁ = m(x − x₁)

where:

• m = slope
• (x₁, y₁) = a known point on the line
• (x, y) = any other point on the line

Point-slope form is especially useful when the known point is not the y-intercept.

---

Understanding Point-Slope Form

Consider a line with:

slope = 2

and a known point:

(3, 5)

Point-slope form allows us to immediately write:

y − 5 = 2(x − 3)

Notice how each piece of information appears directly in the equation:

m = 2

x₁ = 3

y₁ = 5

Therefore:

y − y₁ = m(x − x₁)

becomes:

y − 5 = 2(x − 3)

---

Why Point-Slope Form Works

Slope is defined as:

m = change in y/change in x

Using a known point (x₁, y₁) and another point (x, y):

m = (y − y₁)/(x − x₁)

Multiply both sides by:

(x − x₁)

This gives:

y − y₁ = m(x − x₁)

So point-slope form comes directly from the definition of slope.

---

When Point-Slope Form Is Useful

Point-slope form is particularly useful when a problem gives:

a slope and one point

For example:

Write the equation of a line with slope 4 that passes through (2, 7).

There is no need to find the y-intercept first.

Simply substitute:

m = 4

x₁ = 2

y₁ = 7

Therefore:

y − 7 = 4(x − 2)

This is already a complete equation of the line.

---

Point-Slope Form vs Slope-Intercept Form

Two common forms of a linear equation are:

Point-slope form

y − y₁ = m(x − x₁)

Slope-intercept form

y = mx + b

Point-slope form is usually most convenient when you know:

a point and a slope

Slope-intercept form is usually most convenient when you know:

the slope and y-intercept

Both forms can describe the same line.

---

Identifying the Known Point

Consider:

y − 6 = 3(x − 2)

Compare with:

y − y₁ = m(x − x₁)

We can identify:

m = 3

x₁ = 2

y₁ = 6

Therefore, the line:

• has slope 3
• passes through (2, 6)

---

Negative Coordinates Require Care

Consider:

y + 4 = 2(x − 3)

Point-slope form is:

y − y₁ = m(x − x₁)

Notice:

y + 4 = y − (−4)

Therefore:

y₁ = −4

The known point is:

(3, −4)

and:

m = 2

---

Two Negative Coordinates

Consider:

y + 5 = −3(x + 2)

Rewrite mentally as:

y − (−5) = −3[x − (−2)]

Therefore:

m = −3

and the point is:

(−2, −5)

This is one of the most common places students make sign errors.

---

Writing Point-Slope Form from a Point and Slope

Suppose:

m = 5

point = (4, 7)

Start with:

y − y₁ = m(x − x₁)

Substitute:

y − 7 = 5(x − 4)

That is the equation in point-slope form.

There is no need to simplify unless the question asks for another form.

---

Worked Example: Positive Slope

Write the equation of the line with:

slope = 3

passing through:

(2, 8)

Substitute:

m = 3

x₁ = 2

y₁ = 8

Therefore:

y − 8 = 3(x − 2)

This is the point-slope equation.

---

Worked Example: Negative Slope

Write the equation of a line with:

m = −2

passing through:

(4, 5)

Substitute:

y − 5 = −2(x − 4)

The negative slope means the line falls from left to right.

---

Worked Example: Negative Coordinate

Write the equation of a line with:

m = 4

passing through:

(−3, 2)

Substitute:

y − 2 = 4[x − (−3)]

Subtracting a negative becomes addition:

y − 2 = 4(x + 3)

This is the correct point-slope equation.

---

Worked Example: Both Coordinates Negative

Write the equation of a line with:

m = −1/2

passing through:

(−4, −6)

Substitute:

y − (−6) = −1/2[x − (−4)]

Simplify:

y + 6 = −1/2(x + 4)

The signs come directly from subtracting the coordinates.

---

Point-Slope Form with Zero Slope

Suppose a line has:

m = 0

and passes through:

(3, 7)

Point-slope form gives:

y − 7 = 0(x − 3)

Therefore:

y − 7 = 0

and:

y = 7

This is a horizontal line.

---

Vertical Lines

A vertical line has:

undefined slope

Therefore, the standard point-slope form cannot be used with a numerical value of m for a vertical line.

If a vertical line passes through:

(4, 7)

every point on the line has:

x = 4

Therefore, its equation is:

x = 4

This is an important exception.

---

Graphing from Point-Slope Information

Suppose a line has:

m = 2/3

and passes through:

(1, 2)

Plot the known point:

(1, 2)

Then use:

m = 2/3

which means:

right 3

up 2

This gives another point:

(4, 4)

Repeat the same movement if needed and draw the line.

---

Converting to Slope-Intercept Form

Point-slope form can be converted into:

y = mx + b

The main steps are:

1. distribute the slope
2. simplify
3. isolate y

For example:

y − 5 = 2(x − 3)

Distribute:

y − 5 = 2x − 6

Add 5 to both sides:

y = 2x − 1

Therefore:

point-slope form:

y − 5 = 2(x − 3)

slope-intercept form:

y = 2x − 1

Both equations describe exactly the same line.

---

Worked Conversion Example

Convert:

y − 4 = 3(x + 2)

to slope-intercept form.

Distribute:

y − 4 = 3x + 6

Add 4:

y = 3x + 10

Therefore:

m = 3

b = 10

---

Conversion with a Negative Slope

Convert:

y − 6 = −2(x − 5)

Distribute:

y − 6 = −2x + 10

Add 6:

y = −2x + 16

The slope remains:

−2

Converting the form of an equation does not change the line.

---

Conversion with Negative Coordinates

Convert:

y + 3 = 4(x + 2)

Distribute:

y + 3 = 4x + 8

Subtract 3:

y = 4x + 5

The original point was:

(−2, −3)

Check:

−3 = 4(−2) + 5

−3 = −8 + 5

−3 = −3

Correct.

---

Finding the y-Intercept from Point-Slope Form

Sometimes converting to slope-intercept form reveals useful information.

Consider:

y − 7 = 3(x − 2)

Expand:

y − 7 = 3x − 6

Add 7:

y = 3x + 1

Therefore:

m = 3

b = 1

The y-intercept is:

(0, 1)

---

Finding the Equation from Two Points

Sometimes the slope is not given directly.

Suppose a line passes through:

(2, 3) and (6, 11)

First calculate the slope:

m = (11 − 3)/(6 − 2)

m = 8/4

m = 2

Now choose either known point.

Using (2, 3):

y − 3 = 2(x − 2)

This is the point-slope equation.

---

Either Point Can Be Used

Using the other point:

(6, 11)

we get:

y − 11 = 2(x − 6)

These equations look different:

y − 3 = 2(x − 2)

y − 11 = 2(x − 6)

but they describe:

the same line

---

Verifying That the Equations Are Equivalent

First equation:

y − 3 = 2(x − 2)

Expand:

y − 3 = 2x − 4

y = 2x − 1

Second equation:

y − 11 = 2(x − 6)

Expand:

y − 11 = 2x − 12

y = 2x − 1

Both simplify to the same slope-intercept equation.

---

Determining Whether a Point Lies on a Line

To determine whether a point lies on a line:

1. substitute its x-coordinate for x
2. substitute its y-coordinate for y
3. simplify both sides
4. check whether the equation is true

If the equation is true:

the point lies on the line

If it is false:

the point does not lie on the line

---

Worked Example: Point on the Line

Does:

(4, 9)

lie on:

y − 3 = 2(x − 1)?

Substitute:

9 − 3 = 2(4 − 1)

6 = 2(3)

6 = 6

The statement is true.

Therefore:

(4, 9) lies on the line.

---

Worked Example: Point Not on the Line

Does:

(5, 8)

lie on:

y − 2 = 3(x − 1)?

Substitute:

8 − 2 = 3(5 − 1)

6 = 12

This is false.

Therefore:

(5, 8) does not lie on the line.

---

Why Substitution Works

An equation of a line describes:

every point on that line

Any coordinate pair on the line must make the equation true.

A coordinate pair not on the line will not satisfy the equation.

This idea applies to equations far beyond point-slope form.

---

Checking the Given Point

The point used to create a point-slope equation should always satisfy the equation automatically.

For:

y − 5 = 3(x − 2)

the given point is:

(2, 5)

Substitute:

5 − 5 = 3(2 − 2)

0 = 3(0)

0 = 0

Correct.

This provides a quick check for sign errors.

---

Writing an Equation from a Graph

Suppose a graph clearly shows:

a point at (3, 4)

and slope:

−2

Point-slope form gives:

y − 4 = −2(x − 3)

If slope-intercept form is required:

y − 4 = −2x + 6

y = −2x + 10

Point-slope form can therefore be an efficient intermediate step when working from graphs.

---

Why Point-Slope Form Can Be Faster

Suppose you know:

m = 4

and:

(7, 12)

Using slope-intercept form immediately would require:

12 = 4(7) + b

12 = 28 + b

b = −16

then:

y = 4x − 16

Point-slope form allows:

y − 12 = 4(x − 7)

immediately.

Both are correct, but point-slope form requires fewer initial steps.

---

Real-World Meaning

Point-slope form is useful when you know:

• a rate of change
• one known measurement

For example, suppose the temperature increases steadily at:

2°C per hour

and after 3 hours the temperature is:

18°C

Let:

x = time in hours

y = temperature in °C

Known point:

(3, 18)

Slope:

2°C/hour

Therefore:

y − 18 = 2(x − 3)

This equation models the temperature.

---

Converting the Temperature Model

Start with:

y − 18 = 2(x − 3)

Expand:

y − 18 = 2x − 6

Add 18:

y = 2x + 12

This tells us that the model predicts:

12°C at x = 0

Point-slope form emphasized the known observation.

Slope-intercept form reveals the predicted starting value.

---

Real-World Example: Savings

Suppose a person saves:

$30 per week

After 5 weeks, they have:

$250

Let:

x = weeks

y = savings in dollars

Slope:

m = 30

Known point:

(5, 250)

Point-slope equation:

y − 250 = 30(x − 5)

Convert:

y − 250 = 30x − 150

y = 30x + 100

The model indicates an initial amount of:

$100

---

Real-World Example: Water Level

A reservoir's water level is decreasing at:

0.4 m per day

On day 10, its measured level is:

18 m

Slope:

m = −0.4

Known point:

(10, 18)

Point-slope model:

y − 18 = −0.4(x − 10)

This directly incorporates the measured value and the rate of change.

---

Using the Model to Make a Prediction

Using:

y − 18 = −0.4(x − 10)

find the predicted level on day 15.

Substitute:

x = 15

y − 18 = −0.4(15 − 10)

y − 18 = −0.4(5)

y − 18 = −2

y = 16

The predicted water level is:

16 m

---

Finding an Unknown Point

Suppose a line has slope:

3

and passes through:

(2, 5)

Find y when:

x = 6

Write:

y − 5 = 3(x − 2)

Substitute x = 6:

y − 5 = 3(6 − 2)

y − 5 = 12

y = 17

Therefore, the point:

(6, 17)

lies on the line.

---

Finding an Unknown x-Coordinate

Suppose:

y − 4 = 2(x − 3)

and a point on the line has:

y = 14

Substitute:

14 − 4 = 2(x − 3)

10 = 2(x − 3)

5 = x − 3

x = 8

Therefore:

(8, 14)

lies on the line.

---

Comparing Lines

Suppose:

Line A:

y − 3 = 2(x − 1)

Line B:

y + 5 = 2(x − 4)

Both have:

m = 2

Therefore, the lines have the same slope.

Convert them if needed.

Line A:

y − 3 = 2x − 2

y = 2x + 1

Line B:

y + 5 = 2x − 8

y = 2x − 13

They have different y-intercepts.

Therefore, they are:

parallel

---

Lines Through the Same Point

Consider:

y − 4 = 2(x − 3)

and:

y − 4 = −3(x − 3)

Both lines pass through:

(3, 4)

But their slopes are different.

Therefore, they intersect at:

(3, 4)

Point-slope form makes this relationship particularly easy to see.

---

Connecting Point-Slope and Slope-Intercept Forms

Consider:

y − y₁ = m(x − x₁)

Expand:

y − y₁ = mx − mx₁

Add y₁:

y = mx + (y₁ − mx₁)

Compare with:

y = mx + b

Therefore:

b = y₁ − mx₁

This shows mathematically that point-slope form and slope-intercept form describe the same family of nonvertical straight lines.

---

Choosing the Best Form

Different forms are useful in different situations.

There is no requirement to use only one form forever.

Strong algebra students choose the form that makes the problem easiest.

---

Point-Slope Form and Parallel Lines

Suppose you need a line parallel to:

y = 3x + 7

that passes through:

(4, −2)

Parallel lines have the same slope.

Therefore:

m = 3

Use the new point:

(4, −2)

Point-slope form:

y − (−2) = 3(x − 4)

Therefore:

y + 2 = 3(x − 4)

This is a convenient way to construct a parallel line.

---

Solving a Context Problem

A delivery company knows that the total distance travelled increases at a constant rate of:

60 km/h

After 2 hours, the vehicle has travelled:

135 km

Let:

x = time in hours

y = distance in kilometres

Known point:

(2, 135)

Slope:

60

Equation:

y − 135 = 60(x − 2)

To find the predicted distance at 5 hours:

y − 135 = 60(5 − 2)

y − 135 = 180

y = 315 km

---

Interpreting the Result Carefully

A mathematical model may produce values outside the realistic situation.

For example, extending a constant-speed model backward could produce a mathematically valid y-intercept that does not represent an actual measured starting condition.

Therefore, distinguish between:

what the equation predicts

and

what is meaningful in the real-world context

Always consider the relevant domain.

---

A Reliable Strategy for Writing Point-Slope Form

When given a point and slope:

1. Write y − y₁ = m(x − x₁).
2. Identify m.
3. Identify x₁.
4. Identify y₁.
5. Substitute the values carefully.
6. Keep parentheses around negative coordinates.
7. Simplify double negatives.
8. Check that the known point satisfies the equation.
9. Leave the equation in point-slope form unless another form is requested.

---

A Reliable Strategy for Converting Forms

To convert point-slope form to slope-intercept form:

1. Start with the point-slope equation.
2. Distribute m across the parentheses.
3. Simplify the right side.
4. Add or subtract the appropriate value to isolate y.
5. Write the final equation as y = mx + b.
6. Check that the slope has not changed.
7. Identify the y-intercept.
8. Substitute the original point to verify the equation.

---

A Reliable Strategy for Testing a Point

To determine whether a point lies on a line:

1. Identify the point's x-coordinate.
2. Identify its y-coordinate.
3. Substitute both values into the equation.
4. Simplify the left side.
5. Simplify the right side.
6. Compare the results.
7. If both sides are equal, the point lies on the line.
8. If they are not equal, the point does not lie on the line.

---

Common Mistakes

Using the wrong formula

Point-slope form is:

y − y₁ = m(x − x₁)

Switching x and y coordinates

For (3, 7):

x₁ = 3

y₁ = 7

not the other way around.

Forgetting that subtracting a negative becomes addition

For the point (−4, −2):

y − (−2) = m[x − (−4)]

becomes:

y + 2 = m(x + 4)

Changing the slope during conversion

Converting forms changes the appearance of the equation, not the line itself.

Thinking point-slope form requires the y-intercept

It requires any known point on the line.

Assuming the point in the equation is always positive

The point can contain positive, negative, or zero coordinates.

Testing only the x-coordinate

A point consists of both x and y. Both values must satisfy the equation.

Assuming a vertical line has slope zero

Vertical lines have undefined slope and require equations such as x = 4.

Finding b before using point-slope form

You can do this, but it defeats the main advantage of point-slope form when a point and slope are already known.

---

Key Terms

Point-slope form — linear equation written as y − y₁ = m(x − x₁).

Slope (m) — constant rate of change and measure of a line's steepness and direction.

Known point — coordinate pair (x₁, y₁) known to lie on the line.

Coordinate pair — ordered pair (x, y) representing a point.

Slope-intercept form — linear equation written as y = mx + b.

y-intercept — point where a line crosses the y-axis.

Linear equation — equation whose graph is a straight line.

Rate of change — change in one variable compared with change in another.

Substitution — replacing a variable with a known value.

Equivalent equations — equations written differently that describe the same relationship.

Parallel lines — distinct coplanar lines with equal slopes that never intersect.

Undefined slope — slope of a vertical line.

Domain — set of allowable input values for a mathematical relationship or model.

Linear model — straight-line relationship used to represent a constant rate of change.

---

Key Takeaways

• Point-slope form is y − y₁ = m(x − x₁).
• It is especially useful when one point and the slope are known.
• The point (x₁, y₁) can be any point on the line.
• The known point does not need to be the y-intercept.
• The value m represents the slope.
• Negative coordinates require careful handling because subtracting a negative becomes addition.
• A point and slope determine exactly one nonvertical line.
• Two points can be used to find the slope before writing point-slope form.
• Either of the two known points can be used to write the equation.
• Different-looking point-slope equations can represent the same line.
• Point-slope form can be converted to slope-intercept form by distributing and isolating y.
• Converting forms does not change the slope or graph.
• Substitution can determine whether a point lies on a line.
• A point lies on a line only if its coordinates make the equation true.
• Point-slope form is useful in real-world situations involving a known measurement and a constant rate of change.
• Point-slope form is often convenient for constructing parallel lines because the required slope can be inserted directly.
• Vertical lines cannot be written using ordinary point-slope form because their slope is undefined.
• Choosing the most convenient equation form can make linear problems much easier to solve.

---

Check Your Understanding

1. Write the general equation for point-slope form and explain what m, x₁, and y₁ represent.
2. Write the equation of a line with slope 4 that passes through (3, 7).
3. Write the equation of a line with slope −2 that passes through (5, 1).
4. Write the point-slope equation of a line with slope 3/4 passing through (−2, 5).
5. A line is written as y + 3 = 5(x − 4). Identify its slope and the point used to write the equation.
6. A line is written as y − 6 = −2(x + 3). Identify the slope and the known point.
7. Convert y − 4 = 3(x − 2) into slope-intercept form.
8. Convert y + 5 = −2(x + 1) into slope-intercept form.
9. Determine whether (4, 11) lies on the line y − 3 = 2(x − 1). Show your substitution.
10. Determine whether (5, 7) lies on y + 1 = 3(x − 2).
11. Find the equation of the line passing through (2, 5) and (6, 13). First calculate the slope, then write the equation in point-slope form.
12. A line has slope −3 and passes through (4, 10). Find the value of y when x = 7.
13. A person's savings increase by $25 per week. After 6 weeks, the person has $280. Write a point-slope equation for the savings and then convert it to slope-intercept form.
14. Write an equation for a line parallel to y = 4x − 7 that passes through (3, 2).
15. A student is asked to write the equation of a line with slope 2 passing through (−3, −5) and writes y − 5 = 2(x − 3). Explain both sign errors and write the correct point-slope equation.

Writing Equations of Lines

A linear equation describes a relationship with a constant rate of change. Its graph is a straight line.

There are several ways to write the equation of a line. The best method depends on the information you are given.

You might know:

• two points
• a slope and one point
• the slope and y-intercept
• a graph
• a table of values
• a real-world rate and starting value

The goal is to use the available information to determine an equation that describes every point on the line.

---

Important Forms of a Linear Equation

Two forms are especially useful.

Slope-intercept form

y = mx + b

where:

• m = slope
• b = y-intercept

Point-slope form

y − y₁ = m(x − x₁)

where:

• m = slope
• (x₁, y₁) = known point on the line

Both forms can describe the same line.

---

Choosing an Equation Form

The information given often suggests the easiest form.

Choosing an efficient form can greatly reduce the amount of algebra required.

---

Writing an Equation from Two Points

Suppose a line passes through:

(2, 5) and (6, 13)

We do not initially know the slope or y-intercept.

First calculate the slope:

m = (y₂ − y₁)/(x₂ − x₁)

Substitute:

m = (13 − 5)/(6 − 2)

m = 8/4

m = 2

Now we know the slope.

---

Using Point-Slope Form

Choose either known point.

Using:

(2, 5)

and:

m = 2

substitute into:

y − y₁ = m(x − x₁)

This gives:

y − 5 = 2(x − 2)

This is already a correct equation of the line.

---

Converting to Slope-Intercept Form

Start with:

y − 5 = 2(x − 2)

Distribute:

y − 5 = 2x − 4

Add 5:

y = 2x + 1

Therefore, the same line can be written as:

Point-slope form: y − 5 = 2(x − 2)

Slope-intercept form: y = 2x + 1

---

Checking the Equation

A useful habit is to check the equation using the second point.

Equation:

y = 2x + 1

Second point:

(6, 13)

Substitute:

13 = 2(6) + 1

13 = 12 + 1

13 = 13

The equation is correct.

---

Either Point Can Be Used

Return to:

(2, 5) and (6, 13)

We found:

m = 2

Instead of using (2, 5), use:

(6, 13)

Point-slope form becomes:

y − 13 = 2(x − 6)

Expand:

y − 13 = 2x − 12

y = 2x + 1

Both points produce the same line.

---

Worked Example: Negative Slope

Find the equation through:

(1, 8) and (5, 0)

First find slope:

m = (0 − 8)/(5 − 1)

m = −8/4

m = −2

Use (1, 8):

y − 8 = −2(x − 1)

Expand:

y − 8 = −2x + 2

Add 8:

y = −2x + 10

Check using (5, 0):

0 = −2(5) + 10

0 = 0

Correct.

---

Worked Example with Negative Coordinates

Find the equation through:

(−2, 1) and (4, 13)

Slope:

m = (13 − 1)/(4 − (−2))

m = 12/6

m = 2

Use (−2, 1):

y − 1 = 2[x − (−2)]

y − 1 = 2(x + 2)

Expand:

y − 1 = 2x + 4

y = 2x + 5

---

Writing an Equation from a Graph

A graph usually provides two pieces of information:

• slope
• y-intercept

If both can be identified, use:

y = mx + b

For example, suppose a graph crosses the y-axis at:

(0, 3)

and rises:

2 units for every 1 unit to the right

Then:

m = 2

b = 3

Therefore:

y = 2x + 3

---

Finding Slope from a Graph

Choose two clear points on the line.

Suppose the points are:

(1, 3) and (4, 9)

Calculate:

m = (9 − 3)/(4 − 1)

m = 6/3

m = 2

If the graph crosses the y-axis at:

(0, 1)

then:

b = 1

Therefore:

y = 2x + 1

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Using Rise and Run from a Graph

Slope can also be found directly from a graph using:

m = rise/run

Suppose the line moves:

right 4

up 3

Then:

m = 3/4

If the y-intercept is:

(0, −2)

then:

y = (3/4)x − 2

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Graphs with Negative Slope

Suppose a graph crosses the y-axis at:

(0, 6)

and moves:

right 2

down 3

Then:

rise = −3

run = 2

Therefore:

m = −3/2

The equation is:

y = −(3/2)x + 6

The negative slope matches the fact that the line falls from left to right.

---

When the y-Intercept Is Not Easy to Read

Sometimes the graph does not clearly show the y-intercept.

Suppose you can identify:

m = 3

and the point:

(4, 10)

Instead of trying to estimate the y-intercept, use point-slope form:

y − 10 = 3(x − 4)

Then simplify:

y − 10 = 3x − 12

y = 3x − 2

Now we know:

b = −2

This is often more accurate than estimating an intercept from a graph.

---

Writing an Equation from a Point and Slope

Suppose you know:

m = 4

and the line passes through:

(3, 7)

Point-slope form is the natural choice:

y − 7 = 4(x − 3)

If slope-intercept form is required:

y − 7 = 4x − 12

y = 4x − 5

---

Worked Example: Fractional Slope

A line has:

m = 2/3

and passes through:

(6, 5)

Use point-slope form:

y − 5 = 2/3(x − 6)

Distribute:

y − 5 = 2/3x − 4

Add 5:

y = 2/3x + 1

---

Worked Example: Negative Point

A line has slope:

−3

and passes through:

(−2, 4)

Use point-slope form:

y − 4 = −3[x − (−2)]

Simplify:

y − 4 = −3(x + 2)

Expand:

y − 4 = −3x − 6

y = −3x − 2

---

Finding b Directly

You can also find the y-intercept by substituting a point into:

y = mx + b

Suppose:

m = 3

and:

(4, 17)

Substitute:

17 = 3(4) + b

17 = 12 + b

b = 5

Therefore:

y = 3x + 5

This method is especially useful when the final answer must be in slope-intercept form.

---

Point-Slope or Slope-Intercept?

Suppose you are given:

m = −2

point = (7, 5)

Point-slope form gives the equation immediately:

y − 5 = −2(x − 7)

If the question only asks for an equation, you may be finished.

If the question specifically asks for slope-intercept form, continue:

y − 5 = −2x + 14

y = −2x + 19

The most appropriate form depends on the purpose of the problem.

---

Writing an Equation from a Table

Tables can also represent linear relationships.

Consider:

Each time x increases by 1:

y increases by 3.

Therefore:

m = 3

When x = 0:

y = 4

Therefore:

b = 4

The equation is:

y = 3x + 4

---

A Table Without x = 0

Consider:

Calculate the slope using two rows:

m = (18 − 9)/(5 − 2)

m = 9/3

m = 3

Now use:

(2, 9)

in:

y = mx + b

9 = 3(2) + b

9 = 6 + b

b = 3

Therefore:

y = 3x + 3

---

Checking for a Constant Rate of Change

Before writing a linear equation from data, make sure the relationship is actually linear.

Consider:

Each time:

Δx = 1

and:

Δy = 3

Therefore:

Δy/Δx = 3

throughout the table.

The rate of change is constant, so a linear model is appropriate.

---

Unequal x-Intervals

Do not compare only the changes in y when x-values are unevenly spaced.

Consider:

First interval:

m = (11 − 5)/(3 − 1)

m = 6/2

m = 3

Second interval:

m = (23 − 11)/(7 − 3)

m = 12/4

m = 3

The rate of change remains constant.

Therefore, the data are linear.

---

Special Case: Horizontal Lines

Suppose two points are:

(2, 5) and (8, 5)

Slope:

m = (5 − 5)/(8 − 2)

m = 0

Therefore, the line is horizontal.

Its equation is:

y = 5

This can also be written:

y = 0x + 5

---

Special Case: Vertical Lines

Suppose two points are:

(4, 2) and (4, 9)

The x-coordinate remains:

4

The slope calculation would require division by zero.

Therefore, the slope is undefined.

The equation is:

x = 4

A vertical line cannot be written in ordinary slope-intercept form because y is not a function of x.

---

Recognizing Which Variable Depends on the Other

In a real-world model, decide which variable is the input and which is the output.

Usually:

x = independent variable

y = dependent variable

For example:

hours worked → input

money earned → output

Then:

money earned depends on hours worked

The equation might be:

y = 18x + 25

where:

18 = dollars earned per hour

25 = starting amount

---

Modeling a Starting Value and Rate

Many real-world linear situations follow:

output = rate × input + starting value

This has exactly the same structure as:

y = mx + b

where:

m = rate of change

b = initial value

This connection makes slope-intercept form particularly useful for modeling.

---

Real-World Example: Taxi Fare

A taxi company charges:

$5 starting fee

plus:

$2.50 per kilometre

Let:

x = distance travelled in kilometres

y = total cost in dollars

Rate:

m = 2.50

Starting value:

b = 5

Therefore:

y = 2.5x + 5

Interpretation:

• slope = $2.50/km
• y-intercept = $5 initial charge

---

Using the Taxi Model

Suppose the trip is:

12 km

Use:

y = 2.5x + 5

Substitute:

y = 2.5(12) + 5

y = 30 + 5

y = $35

The equation allows us to make predictions within the model.

---

Real-World Example: Savings

A student has:

$120

and saves:

$35 each week

Let:

x = weeks

y = savings

Then:

m = 35

b = 120

Equation:

y = 35x + 120

After 8 weeks:

y = 35(8) + 120

y = 280 + 120

y = $400

---

Real-World Example: Decreasing Quantity

A tank initially contains:

900 L

Water drains at:

25 L/min

Let:

x = time in minutes

y = water remaining

Because the amount decreases:

m = −25

Initial amount:

b = 900

Therefore:

y = −25x + 900

The negative slope represents the decrease.

---

Modeling from a Known Measurement

Sometimes the initial value is unknown.

Suppose a plant grows at a constant modeled rate of:

2.5 cm/week

At week 4, its height is:

23 cm

Known point:

(4, 23)

Slope:

m = 2.5

Point-slope form is convenient:

y − 23 = 2.5(x − 4)

Simplify:

y − 23 = 2.5x − 10

y = 2.5x + 13

The model predicts an initial height of:

13 cm

---

Real Measurements and Linear Models

Real data do not always fall exactly on a straight line.

For example, plant growth, temperature, sales, and travel may change at different rates.

A linear equation is appropriate when:

• the rate is approximately constant
• a straight line reasonably represents the data
• the model is used over an appropriate range

An equation may be mathematically valid for every x-value but unrealistic outside the context.

---

Interpolation

Interpolation means using a model to estimate a value between known data values.

Suppose measurements were collected at:

x = 2 and x = 10

Using the model to estimate a value at:

x = 6

is interpolation.

This is often reasonably reliable when the linear relationship is appropriate.

---

Extrapolation

Extrapolation means using a model to predict beyond the known data range.

If measurements were collected from:

x = 2 to x = 10

using the equation to predict:

x = 50

is extrapolation.

This can be less reliable because the relationship may not remain linear that far beyond the observed data.

---

Comparing Different Equation Forms

Consider the line:

y = 3x − 4

The same line could be written using the point (2, 2):

y − 2 = 3(x − 2)

These equations look different but represent the same line.

Slope-intercept form makes it easy to see:

• slope
• y-intercept

Point-slope form makes it easy to see:

• slope
• a known point

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Standard Form

Another common linear form is:

Ax + By = C

For example:

2x + 3y = 12

This is called standard form.

To reveal the slope and y-intercept, rearrange:

3y = −2x + 12

Divide by 3:

y = −(2/3)x + 4

Now we can see:

m = −2/3

b = 4

Different equation forms emphasize different features of a line.

---

Selecting the Most Appropriate Form

Suppose you know:

slope = 4

and:

y-intercept = −2

The easiest form is:

y = 4x − 2

Suppose instead you know:

slope = 4

and the line passes through:

(7, 10)

The easiest starting form is:

y − 10 = 4(x − 7)

Choosing the form that matches the available information reduces unnecessary calculations.

---

Writing a Parallel Line

Suppose you need a line parallel to:

y = 3x + 5

passing through:

(4, 1)

Parallel lines have equal slopes.

Therefore:

m = 3

Use point-slope form:

y − 1 = 3(x − 4)

Simplify:

y − 1 = 3x − 12

y = 3x − 11

---

Writing a Perpendicular Line

Suppose you need a line perpendicular to:

y = 2x + 4

passing through:

(6, 3)

Original slope:

m = 2

Perpendicular slope:

m = −1/2

Use point-slope form:

y − 3 = −1/2(x − 6)

Simplify:

y − 3 = −1/2x + 3

y = −1/2x + 6

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Determining Whether a Point Fits an Equation

Suppose the equation is:

y = 3x + 2

Does:

(4, 14)

lie on the line?

Substitute:

14 = 3(4) + 2

14 = 14

Yes.

Now test:

(5, 14)

14 = 3(5) + 2

14 = 17

False.

Therefore:

(5, 14) does not lie on the line.

---

Finding a Missing Coordinate

Suppose:

y = 4x − 3

and a point on the line has:

x = 5

Then:

y = 4(5) − 3

y = 20 − 3

y = 17

Therefore:

(5, 17)

lies on the line.

---

Finding x from an Equation

Suppose:

y = 3x + 5

and:

y = 26

Substitute:

26 = 3x + 5

21 = 3x

x = 7

Therefore:

(7, 26)

lies on the line.

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Checking an Equation Using Multiple Representations

A strong way to verify a linear equation is to compare several representations.

For example, suppose you find:

y = 2x + 3

Check:

Slope

Does the line rise 2 for every 1 right?

y-intercept

Does it cross at (0, 3)?

Known points

Do the given coordinates satisfy the equation?

Table

Do calculated values match the data?

If all of these agree, the equation is likely correct.

---

A Complete Two-Point Problem

Find the equation through:

(−3, 8) and (5, −4)

First calculate slope:

m = [−4 − 8]/[5 − (−3)]

m = −12/8

m = −3/2

Use (−3, 8):

y − 8 = −3/2[x − (−3)]

y − 8 = −3/2(x + 3)

Convert:

y − 8 = −3/2x − 9/2

Add 8:

y = −3/2x − 9/2 + 16/2

y = −3/2x + 7/2

---

Checking the Complete Example

Use:

(5, −4)

Equation:

y = −3/2x + 7/2

Substitute:

−4 = −3/2(5) + 7/2

−4 = −15/2 + 7/2

−4 = −8/2

−4 = −4

Correct.

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A Complete Real-World Model

A streaming service charges a fixed monthly fee of:

$12

plus:

$3 for each premium rental

Let:

x = number of premium rentals

y = total monthly cost

The fixed fee represents:

b = 12

The cost per rental represents:

m = 3

Therefore:

y = 3x + 12

For 6 rentals:

y = 3(6) + 12

y = 18 + 12

y = $30

The equation provides both a description of the relationship and a way to make predictions.

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Domain Restrictions in Context

The equation:

y = 3x + 12

mathematically allows values such as:

x = −4

or:

x = 2.7

But if x represents the number of premium rentals, those values may not make sense.

The realistic domain might be:

x = 0, 1, 2, 3, ...

Therefore, mathematical models must be interpreted according to their context.

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Discrete and Continuous Models

Some linear situations are continuous.

Examples:

• time
• distance
• temperature
• volume

Values between measurements may make sense.

Other situations are discrete.

Examples:

• number of tickets
• number of students
• number of products
• number of rentals

A line equation may describe the relationship algebraically, but only certain points may represent realistic possibilities.

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A Reliable Strategy: Two Points

When given two points:

1. Label the points.
2. Calculate slope using m = (y₂ − y₁)/(x₂ − x₁).
3. Simplify the slope.
4. Choose either point.
5. Substitute into point-slope form.
6. Write the equation.
7. Convert to slope-intercept form if required.
8. Check the equation using the other point.

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A Reliable Strategy: From a Graph

When given a graph:

1. Identify two clear points.
2. Calculate rise and run.
3. Determine the slope.
4. Find the y-intercept if visible.
5. Write y = mx + b.
6. If the y-intercept is unclear, use a known point and point-slope form.
7. Check that the slope sign matches the graph's direction.
8. Verify another point.

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A Reliable Strategy: Point and Slope

When given a point and slope:

1. Identify m.
2. Identify (x₁, y₁).
3. Write y − y₁ = m(x − x₁).
4. Substitute carefully.
5. Simplify double negatives.
6. Leave in point-slope form if appropriate.
7. Convert to y = mx + b if required.
8. Check the original point.

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A Reliable Strategy: Real-World Models

When writing a linear model:

1. Identify the input variable.
2. Identify the output variable.
3. Decide whether a linear model is appropriate.
4. Identify the rate of change.
5. Determine whether an initial value is known.
6. Choose a useful equation form.
7. Write the equation.
8. Include meaningful units.
9. Use the equation to calculate or predict values.
10. Interpret the result in context.
11. Consider whether the input is discrete or continuous.
12. Consider the realistic domain of the model.

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Common Mistakes

Finding b before finding the slope when given two points

Usually find the slope first.

Using the slope formula backward

Slope is:

change in y/change in x

Changing subtraction order halfway through

If you calculate y₂ − y₁, also calculate x₂ − x₁.

Forgetting negative coordinates

Use parentheses when substituting negative values.

Assuming the y-intercept must be visible

If it is not, use a known point and the slope.

Choosing slope-intercept form when point-slope form is much easier

Use the form that matches the information given.

Forgetting that horizontal lines have slope zero

Their equations have the form:

y = constant

Trying to write a vertical line as y = mx + b

Vertical lines have equations such as:

x = 4

Ignoring units in real-world problems

A slope such as 5 is much more meaningful as 5 km/h, $5/item, or 5 L/min.

Assuming every data set is linear

Check whether the rate of change is reasonably constant.

Using a model outside a realistic domain

A mathematically valid answer may not make sense physically.

Treating discrete data as continuous

If x represents people or products, fractional values may not be meaningful.

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Key Terms

Linear equation — equation whose graph is a straight line.

Slope — constant rate of change of a nonvertical linear relationship.

y-intercept — point where a graph crosses the y-axis.

Slope-intercept form — equation written as y = mx + b.

Point-slope form — equation written as y − y₁ = m(x − x₁).

Standard form — linear equation commonly written as Ax + By = C.

Rate of change — change in output compared with change in input.

Initial value — value of the output when the input is zero, when meaningful.

Linear model — linear equation used to represent a real relationship.

Interpolation — estimating within the range of known data.

Extrapolation — predicting beyond the range of known data.

Independent variable — input variable.

Dependent variable — output variable whose value depends on the input.

Domain — set of allowable input values.

Discrete data — values that occur as separate countable possibilities.

Continuous data — values that can vary through an interval.

Equivalent equations — different equations representing the same relationship or line.

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Key Takeaways

• A straight line can be represented by a linear equation.
• Different information may make different equation forms more convenient.
• When given two points, first calculate the slope.
• After finding the slope, point-slope form provides a direct way to write the equation.
• Either of two known points can be used in point-slope form.
• Slope-intercept form is useful when the slope and y-intercept are known.
• A graph can provide slope through rise/run and the y-intercept through its crossing of the y-axis.
• If the y-intercept cannot be read accurately, use a known point instead.
• A table represents a linear relationship when its rate of change is constant.
• Unequal x-intervals require comparing Δy/Δx rather than simply comparing changes in y.
• Horizontal lines have equations of the form y = constant.
• Vertical lines have equations of the form x = constant.
• Point-slope and slope-intercept forms can represent the same line.
• Real-world linear models often have the structure output = rate × input + starting value.
• Slope should be interpreted with units in real-world models.
• The y-intercept often represents an initial value, but x = 0 must make sense in context.
• Interpolation estimates values within known data.
• Extrapolation predicts beyond known data and can be less reliable.
• Real-world models should be used only over reasonable domains.
• Discrete and continuous variables should be distinguished.
• Substitution provides a useful way to verify an equation.
• Selecting the most appropriate equation form makes solving linear problems faster and less error-prone.

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Check Your Understanding

1. Describe when slope-intercept form is more convenient than point-slope form.
2. Find the equation of the line passing through (2, 4) and (6, 12).
3. Find the equation of the line passing through (−3, 2) and (5, 18).
4. A line has slope −4 and passes through (3, 7). Write its equation in point-slope form and slope-intercept form.
5. A graph crosses the y-axis at (0, −3) and rises 5 units for every 2 units moved right. Write its equation.
6. A line passes through (4, 9) and has slope 3/2. Determine its y-intercept and write its equation in slope-intercept form.
7. Determine the equation of the horizontal line passing through (5, −7).
8. Determine the equation of the vertical line passing through (−4, 6).
9. A table contains the points (2, 7), (5, 16), and (9, 28). Determine whether the relationship is linear. If it is, write its equation.
10. Explain why the equation of a line can be written in different forms without changing its graph.
11. A taxi charges a $6 starting fee and $2.25 per kilometre. Write an equation for the total cost y of travelling x kilometres. Explain the meaning of the slope and y-intercept.
12. A tank contains 700 L of water and loses 15 L each minute. Write a linear model and use it to determine the amount remaining after 20 minutes.
13. A quantity increases at 8 units per hour. After 5 hours its value is 62. Write a point-slope equation, convert it to slope-intercept form, and determine its modeled value at x = 0.
14. Explain why extrapolating a real-world linear model far beyond the range of measured data may produce unreliable predictions.
15. A student is given the points (−2, 5) and (4, 17) and writes y = 3x + 11. Determine whether the equation is correct. If it is not, identify the error and write the correct equation.