Slope-Intercept Form

A linear equation represents a straight-line relationship between two variables. One of the most useful ways to write a linear equation is slope-intercept form:

y = mx + b

where:

• m = slope
• b = y-intercept
• x and y = variables

Slope-intercept form is especially useful because we can identify two important features of a line simply by looking at its equation.

For example:

y = 3x + 2

has:

slope = 3

y-intercept = 2

The line crosses the y-axis at:

(0, 2)

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Understanding y = mx + b

The equation:

y = mx + b

contains two numbers that control the position and direction of the line.

m Controls the Slope

The value of m tells us:

• how steep the line is
• whether the line rises or falls
• how much y changes when x changes

b Controls the y-Intercept

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

Because every point on the y-axis has:

x = 0

substituting x = 0 gives:

y = m(0) + b

so:

y = b

Therefore, the y-intercept is:

(0, b)

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Identifying Slope and y-Intercept

Consider:

y = 4x + 7

Compare it with:

y = mx + b

Therefore:

m = 4

b = 7

So:

slope = 4

y-intercept = (0, 7)

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Another Example

Consider:

y = −2x + 5

The slope is:

m = −2

The y-intercept is:

b = 5

Therefore:

y-intercept = (0, 5)

The negative slope tells us that the line falls from left to right.

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What Is Slope?

Slope measures the rate of change of a line.

It compares vertical change with horizontal change:

slope = rise/run

or:

m = change in y / change in x

Using two points:

m = (y₂ − y₁)/(x₂ − x₁)

For example, if y increases by 6 while x increases by 2:

m = 6/2

m = 3

The slope is 3.

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Slope as Rise Over Run

Suppose:

m = 2/3

This means:

rise = 2

run = 3

Starting at a point on the line:

• move 3 units right
• move 2 units up

to find another point.

Slope can also be interpreted in the opposite direction:

• move 3 units left
• move 2 units down

Both movements stay on the same line.

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Whole-Number Slopes

A whole number can always be written as a fraction over 1.

For example:

m = 4

means:

m = 4/1

So from one point:

• move 1 right
• move 4 up

Similarly:

m = −3

can be written:

m = −3/1

so we could move:

• 1 right
• 3 down

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Positive Slope

If:

m > 0

the line rises from left to right.

Example:

y = 2x + 1

As x increases, y increases.

Positive slope represents an increasing relationship.

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Negative Slope

If:

m < 0

the line falls from left to right.

Example:

y = −3x + 4

As x increases, y decreases.

Negative slope represents a decreasing relationship.

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Zero Slope

Consider:

y = 5

This can be written:

y = 0x + 5

Therefore:

m = 0

The graph is a horizontal line.

Its y-value never changes.

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Undefined Slope

A vertical line such as:

x = 4

has undefined slope.

There is no horizontal change:

run = 0

and division by zero is undefined.

A vertical line cannot be written in ordinary slope-intercept form:

y = mx + b

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Types of Slope

Recognizing these patterns helps when interpreting graphs.

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Understanding the y-Intercept

The y-intercept is where the line crosses the y-axis.

For:

y = 3x − 4

the y-intercept is:

b = −4

Therefore the line crosses the y-axis at:

(0, −4)

Notice the difference between:

b = −4

and:

y-intercept = (0, −4)

One is a value; the other is a coordinate.

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Graphing from Slope-Intercept Form

Suppose:

y = 2x + 1

We know:

m = 2

b = 1

To graph the equation:

1. Plot the y-intercept (0, 1).
2. Write the slope as 2/1.
3. From (0, 1), move 1 right and 2 up.
4. Plot the new point.
5. Repeat if useful.
6. Draw a straight line through the points.

The equation itself gives almost everything needed to graph the line.

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Graphing a Fractional Slope

Graph:

y = (2/3)x − 1

Identify:

m = 2/3

b = −1

Start at:

(0, −1)

Then use the slope:

rise = 2

run = 3

Move:

• 3 right
• 2 up

This gives another point:

(3, 1)

Continue the pattern and draw the line.

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Graphing a Negative Slope

Graph:

y = −2x + 4

Start at:

(0, 4)

Write:

−2 = −2/1

From the y-intercept:

• move 1 right
• move 2 down

This produces another point:

(1, 2)

Repeat:

(2, 0)

The line falls from left to right.

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Negative Fractional Slopes

Suppose:

m = −3/4

There are two convenient ways to represent this:

rise = −3, run = 4

or:

rise = 3, run = −4

So you can move:

• 4 right and 3 down

or:

• 4 left and 3 up

Both movements describe the same slope.

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Creating a Table to Graph a Line

Another way to graph:

y = 2x + 1

is to choose several x-values.

Plot these ordered pairs and connect them with a straight line.

This method works, but using the slope and y-intercept is often faster.

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Writing an Equation from a Graph

To write an equation from a graph, identify:

1. the y-intercept
2. the slope

Then substitute them into:

y = mx + b

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

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Finding Slope from Two Points on a Graph

Suppose two points on a line are:

(1, 3)

and:

(4, 9)

Use:

m = (y₂ − y₁)/(x₂ − x₁)

Substitute:

m = (9 − 3)/(4 − 1)

m = 6/3

m = 2

If the graph shows the y-intercept is:

(0, 1)

then:

b = 1

Therefore:

y = 2x + 1

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Be Consistent When Calculating Slope

Using:

(1, 3) and (4, 9)

we wrote:

m = (9 − 3)/(4 − 1)

If we reverse the order, we must reverse it in both numerator and denominator:

m = (3 − 9)/(1 − 4)

m = −6/−3

m = 2

Both methods give the same result.

Do not reverse only one part.

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Writing an Equation from Slope and y-Intercept

Suppose:

slope = 5

and:

y-intercept = −3

Use:

y = mx + b

Substitute:

m = 5

b = −3

Therefore:

y = 5x − 3

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Writing an Equation from Slope and a Point

Sometimes the slope is known, but the y-intercept is not.

Suppose:

m = 3

and the line passes through:

(2, 7)

Start with:

y = mx + b

Substitute:

x = 2

y = 7

m = 3

giving:

7 = 3(2) + b

7 = 6 + b

Therefore:

b = 1

So the equation is:

y = 3x + 1

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Why This Method Works

The point:

(2, 7)

lies on the line.

Therefore its x- and y-values must satisfy the equation.

We know:

y = mx + b

and:

m = 3

so:

y = 3x + b

Using the known point lets us determine the only missing value, b.

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Another Slope-and-Point Example

A line has:

m = −2

and passes through:

(3, 1)

Start:

y = mx + b

Substitute:

1 = −2(3) + b

1 = −6 + b

Add 6:

b = 7

Therefore:

y = −2x + 7

Check the point:

1 = −2(3) + 7

1 = 1 ✓

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Finding an Equation from Two Points

Suppose a line passes through:

(2, 5)

and:

(6, 13)

First find the slope:

m = (13 − 5)/(6 − 2)

m = 8/4

m = 2

Now use one point to find b.

Using:

(2, 5)

in:

y = mx + b

gives:

5 = 2(2) + b

5 = 4 + b

b = 1

Therefore:

y = 2x + 1

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Verify with the Second Point

Use:

(6, 13)

in:

y = 2x + 1

13 = 2(6) + 1

13 = 13 ✓

The second point confirms that the equation is correct.

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Converting to Slope-Intercept Form

Not every equation starts in the form:

y = mx + b

Consider:

2x + y = 7

Solve for y:

y = −2x + 7

Now we can immediately identify:

m = −2

b = 7

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Another Conversion Example

Consider:

3x + 2y = 10

Solve for y:

2y = −3x + 10

Divide everything by 2:

y = −(3/2)x + 5

Therefore:

m = −3/2

b = 5

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Be Careful When Dividing

For:

4x + 2y = 12

subtract 4x:

2y = −4x + 12

Divide every term by 2:

y = −2x + 6

Not:

y = −4x + 6

Both terms on the right must be divided.

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Slope as Rate of Change

Slope is much more than the steepness of a graph.

It describes how one quantity changes relative to another.

Suppose:

C = 3x + 10

where:

• C = total cost
• x = number of items

The slope is:

3

This means the cost increases by:

$3 per item

The intercept is:

10

This represents a starting cost of:

$10

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Interpreting the Intercept in Context

In:

C = 3x + 10

when:

x = 0

we have:

C = 10

So the intercept represents the value before any items are purchased.

Depending on the context, this could represent:

• an initial fee
• a starting amount
• a fixed cost
• a base charge
• an initial measurement

The meaning depends on what x and y represent.

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Taxi Fare Example

Suppose:

C = 2.5d + 4

where:

• C = cost in dollars
• d = distance in kilometres

The slope:

2.5

means:

$2.50 per kilometre

The intercept:

4

means:

$4 initial charge

If the trip is 6 km:

C = 2.5(6) + 4

C = 15 + 4

C = $19

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Savings Example

Suppose:

S = 25w + 100

where:

• S = total savings
• w = number of weeks

The slope is:

25

This means savings increase by:

$25 per week

The intercept is:

100

This means the person started with:

$100

After 8 weeks:

S = 25(8) + 100

S = 300

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Temperature Example

Suppose the temperature of an object is modeled by:

T = −3t + 90

where:

• T = temperature in °C
• t = time in minutes

The slope:

−3

means the temperature decreases by:

3°C per minute

The intercept:

90

means the initial temperature at t = 0 is:

90°C

The negative slope represents cooling.

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Distance Example

Suppose:

d = 60t + 20

where:

• d = distance from a reference point in kilometres
• t = time in hours

The slope:

60

means the object travels away from the reference point at:

60 km/h

The intercept:

20

means the object began:

20 km from the reference point

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Science Example

Suppose an experiment produces:

V = 2.4t + 1.5

where:

• V = volume in mL
• t = time in seconds

The slope:

2.4 mL/s

represents the rate at which volume increases.

The intercept:

1.5 mL

represents the initial volume at:

t = 0

The units of slope are:

units of y / units of x

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Units of Slope

Since:

slope = change in y / change in x

the units of slope come from dividing the units of y by the units of x.

Examples:

Distance vs time:

km/h

Cost vs number of items:

$/item

Temperature vs time:

°C/min

Mass vs volume:

g/cm³

Population vs time:

people/year

Units help explain what slope actually means.

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Comparing Slopes

Consider:

Line A: y = 2x + 1

Line B: y = 5x + 1

Both have the same y-intercept.

But:

m_A = 2

m_B = 5

Line B increases faster and is steeper.

For positive slopes, a larger slope means a faster increase.

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Comparing Negative Slopes

Consider:

Line A: y = −2x + 4

Line B: y = −5x + 4

Both decrease.

Line B has the larger magnitude of slope:

|−5| > |−2|

so Line B falls more steeply.

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Parallel Lines

Consider:

y = 3x + 2

y = 3x − 5

Both have:

m = 3

but different y-intercepts.

Therefore the lines have the same steepness and never meet.

They are parallel.

For different nonvertical lines:

same slope + different intercepts → parallel lines

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The Same Line

Consider:

y = 2x + 4

and:

2y = 4x + 8

Divide the second equation by 2:

y = 2x + 4

The equations are identical.

They represent the same line.

This idea becomes particularly important when solving systems of equations.

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Connection to Systems of Equations

Slope-intercept form can help predict the number of solutions in a system.

If two lines have:

different slopes

they intersect once.

Therefore:

one solution

If they have:

same slope, different y-intercepts

they are parallel.

Therefore:

no solution

If they have:

same slope, same y-intercept

they are the same line.

Therefore:

infinitely many solutions

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A Useful Comparison

This lets us sometimes classify a system without fully solving it.

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Writing Equations from Real Situations

Suppose a gym charges:

• $25 membership fee
• $8 per visit

Let:

C = total cost

v = number of visits

The fixed starting amount becomes the intercept:

b = 25

The cost per visit becomes the slope:

m = 8

Therefore:

C = 8v + 25

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Identifying Slope and Intercept from Words

Look for language such as:

per, each, every, for each

These often indicate slope.

Look for:

starting, initial, base, fixed, at zero

These often indicate the intercept.

For example:

A tank begins with 50 L of water and fills at 4 L per minute.

Equation:

V = 4t + 50

Slope:

4 L/min

Intercept:

50 L

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When the Intercept Is Zero

Suppose:

d = 70t

This can be written:

d = 70t + 0

Therefore:

m = 70

b = 0

The graph passes through:

(0, 0)

This often represents a direct proportional relationship.

For example, an object starting at zero distance and travelling at constant speed may have:

d = vt

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When the Slope Is Zero

Suppose:

T = 25

This can be written:

T = 0t + 25

Therefore:

m = 0

The temperature remains constant at:

25°C

The graph is horizontal.

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Recognizing Hidden Coefficients

Consider:

y = x + 6

There is an invisible coefficient of 1:

y = 1x + 6

Therefore:

m = 1

b = 6

For:

y = −x + 4

the coefficient is:

−1

Therefore:

m = −1

b = 4

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No Written Constant

Consider:

y = 5x

This means:

y = 5x + 0

Therefore:

m = 5

b = 0

The line passes through the origin.

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Writing a Complete Interpretation

Suppose:

C = 12t + 40

where C is cost in dollars and t is time in hours.

A weak interpretation is:

The slope is 12 and the intercept is 40.

A stronger interpretation is:

The cost increases by $12 for each additional hour, and the initial cost at 0 hours is $40.

Good mathematical communication explains what the numbers mean in context.

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A Reliable Strategy

When working with slope-intercept form, use:

Identify → Interpret → Graph or Construct → Check

Identify

Find m and b.

Interpret

Determine what the slope and intercept mean.

Graph or Construct

Use m and b to graph the line, or use known information to create the equation.

Check

Confirm that known points satisfy the equation and that the graph or equation is reasonable.

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Common Mistakes

Mistake: Switching m and b.

In:

y = mx + b

m is the slope and b is the y-intercept.

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Mistake: Ignoring the sign of the slope.

In:

y = −3x + 2

the slope is:

−3

not 3.

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Mistake: Saying the y-intercept is just a point without the zero x-coordinate.

If:

b = 5

the y-intercept is:

(0, 5)

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Mistake: Reading the constant as the slope.

For:

y = 2x + 7

slope = 2, not 7.

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Mistake: Treating 3 as though it cannot be a fraction.

3 = 3/1

so its rise/run interpretation is:

3 up, 1 right

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Mistake: Using inconsistent subtraction when finding slope.

If you calculate:

y₂ − y₁

you must also calculate:

x₂ − x₁

in the same order.

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Mistake: Forgetting that a negative slope can be shown in different ways.

−3/4 = 3/(−4)

Both represent the same slope.

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Mistake: Forgetting to divide every term when rearranging into slope-intercept form.

From:

2y = −6x + 10

the correct result is:

y = −3x + 5

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Mistake: Assuming the y-intercept is always positive.

The line may cross below the origin.

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Mistake: Ignoring units in real-world problems.

A slope of 5 might mean:

• $5/item
• 5 km/h
• 5°C/min
• 5 L/s

depending on the situation.

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Key Terms

Linear equation — equation whose graph is a straight line.

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

Slope — rate of change of y relative to x.

Rate of change — amount one quantity changes compared with another.

Rise — vertical change between two points.

Run — horizontal change between two points.

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

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

Positive slope — slope describing a line that rises from left to right.

Negative slope — slope describing a line that falls from left to right.

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line.

Slope formula — m = (y₂ − y₁)/(x₂ − x₁).

Coordinate — ordered pair describing a point's position.

Origin — point (0, 0).

Slope-intercept equation — equation in which slope and y-intercept can be read directly.

Rate — comparison between quantities with different units.

Initial value — value of y when x = 0.

Parallel lines — lines in the same plane with equal slopes that never intersect.

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Key Takeaways

• Slope-intercept form is y = mx + b.
• The coefficient m represents the slope.
• The constant b represents the y-intercept.
• The y-intercept occurs at (0, b).
• Slope measures rate of change.
• Slope can be calculated using rise/run.
• Using two points, m = (y₂ − y₁)/(x₂ − x₁).
• Positive slopes rise from left to right.
• Negative slopes fall from left to right.
• A horizontal line has slope 0.
• A vertical line has undefined slope.
• A whole-number slope can be written over 1.
• To graph from slope-intercept form, plot the y-intercept first and then use the slope.
• A line's equation can be written from its graph by determining m and b.
• If the slope and one point are known, substitute the point into y = mx + b to find b.
• Two points can be used to calculate the slope and then determine the equation.
• Equations can often be rearranged into slope-intercept form by solving for y.
• In real situations, slope represents a rate of change.
• The units of slope are units of y divided by units of x.
• The y-intercept often represents an initial or starting value.
• A zero y-intercept means the line passes through the origin.
• Lines with the same slope and different intercepts are parallel.
• Lines with different slopes intersect once.
• Lines with the same slope and same intercept are the same line.
• Slope-intercept form is therefore especially useful when graphing and analyzing systems of linear equations.

Check Your Understanding

1. State the general slope-intercept form of a linear equation.
2. What does m represent in y = mx + b?
3. What does b represent?
4. Identify the slope and y-intercept of y = 4x + 7.
5. Identify the slope and y-intercept of y = −3x + 5.
6. Identify the slope and y-intercept of y = x − 6.
7. Identify the slope and y-intercept of y = −x + 8.
8. Identify the slope and y-intercept of y = 5x.
9. Describe how you would graph y = 2x + 3 using the slope and y-intercept.
10. Describe how you would graph y = −3x + 4.
11. Find the slope of the line through (2, 3) and (6, 11).
12. Find the slope of the line through (−1, 5) and (3, −3).
13. A graph has slope 3 and y-intercept −2. Write its equation.
14. A line has slope 4 and passes through (2, 11). Find its equation in slope-intercept form.
15. A line has slope −2 and passes through (3, 1). Find its equation.
16. Find the equation of the line passing through (2, 5) and (6, 13).
17. Rewrite 3x + 2y = 10 in slope-intercept form and identify its slope and y-intercept.
18. A taxi fare is modeled by C = 2.5d + 4. Explain what both 2.5 and 4 mean in context.
19. A tank contains 80 L of water and fills at 6 L/min. Write a linear equation relating volume V to time t, then identify and interpret its slope and intercept.
20. Two equations are y = 3x + 2 and y = 3x − 5. Without graphing or solving algebraically, explain the relationship between their graphs and determine how many solutions the corresponding system has.