Slope-Intercept Form

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

y = mx + b

where:

• m = slope
• b = y-intercept
• x = independent variable
• y = dependent variable

The slope tells us the rate of change. The y-intercept tells us the value of y when x = 0.

For example:

y = 3x + 5

has:

• slope = 3
• y-intercept = 5

This means the line crosses the y-axis at (0, 5) and rises 3 units for every 1 unit it moves to the right.

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Understanding y = mx + b

The equation:

y = mx + b

contains two pieces of information that completely determine a non-vertical straight line.

The value m determines:

• direction
• steepness
• rate of change

The value b determines:

• where the line crosses the y-axis
• the starting value when x = 0

For:

y = 4x − 2

m = 4

b = −2

Therefore, the line has:

slope = 4

and:

y-intercept = (0, −2)

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Identifying the Slope

In slope-intercept form, the slope is the coefficient of x.

Consider:

y = 7x + 3

The coefficient of x is:

7

Therefore:

m = 7

The line rises 7 units for every increase of 1 unit in x.

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Identifying a Negative Slope

Consider:

y = −3x + 8

The coefficient of x is:

−3

Therefore:

m = −3

The line decreases by 3 units for every increase of 1 unit in x.

The graph falls from left to right.

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Identifying a Fractional Slope

Consider:

y = (2/5)x + 4

The slope is:

m = 2/5

This can be interpreted as:

rise = 2

run = 5

So from one point on the line, move:

• 5 units right
• 2 units up

to find another point.

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Identifying the y-Intercept

The y-intercept is the constant term in:

y = mx + b

Consider:

y = 5x + 9

The y-intercept is:

b = 9

Therefore, the line crosses the y-axis at:

(0, 9)

Remember that the y-intercept is a point on the y-axis, so its coordinate is:

(0, b)

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A Negative y-Intercept

Consider:

y = 2x − 6

This can be viewed as:

y = 2x + (−6)

Therefore:

b = −6

The y-intercept is:

(0, −6)

The line crosses the y-axis below the origin.

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Why b Gives the y-Intercept

Consider:

y = mx + b

At the y-axis:

x = 0

Substitute:

y = m(0) + b

y = b

Therefore, the line crosses the y-axis at:

(0, b)

This explains why b is called the y-intercept.

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Reading Slope and Intercept Quickly

Consider:

y = 6x + 4

m = 6

b = 4

Consider:

y = −2x + 9

m = −2

b = 9

Consider:

y = (1/3)x − 7

m = 1/3

b = −7

Once an equation is in slope-intercept form, these values can usually be identified immediately.

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When the Slope Is Not Written as a Fraction

Any whole number can be written as a fraction over 1.

For:

y = 4x + 2

the slope is:

4 = 4/1

This means:

rise = 4

run = 1

For:

y = −5x + 3

we can write:

−5 = −5/1

This can be interpreted as:

• move 1 right
• move 5 down

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When There Is No Number Written Before x

Consider:

y = x + 4

The coefficient of x is understood to be:

1

Therefore:

m = 1

b = 4

Similarly:

y = −x + 6

means:

y = −1x + 6

Therefore:

m = −1

b = 6

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When There Is No Constant Term

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.

This is also a direct variation.

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Horizontal Lines

Consider:

y = 5

This can be thought of as:

y = 0x + 5

Therefore:

m = 0

b = 5

The graph is a horizontal line through:

(0, 5)

Every point on the line has y = 5.

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Vertical Lines

A vertical line such as:

x = 4

cannot be written in slope-intercept form.

Why?

A vertical line has an undefined slope.

Slope-intercept form:

y = mx + b

requires a defined value of m.

Therefore, vertical lines are an important exception.

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Graphing Using Slope-Intercept Form

Suppose:

y = 2x + 3

Identify:

m = 2

b = 3

First plot the y-intercept:

(0, 3)

Then interpret:

m = 2/1

From (0, 3):

• move 1 unit right
• move 2 units up

This gives:

(1, 5)

Repeat:

(2, 7)

Draw a straight line through the points.

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The Slope Triangle

A slope triangle shows the vertical and horizontal changes between two points.

For:

m = 3/2

we can use:

rise = 3

run = 2

Starting from the y-intercept:

• move 2 units right
• move 3 units up

This creates another point on the line.

The same pattern can be repeated because a linear relationship has a constant slope.

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Graphing a Positive Slope

Consider:

y = 3x − 4

Identify:

m = 3 = 3/1

b = −4

Plot:

(0, −4)

Then move:

• 1 right
• 3 up

This gives:

(1, −1)

Again:

• 1 right
• 3 up

This gives:

(2, 2)

The graph rises from left to right.

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Graphing a Negative Slope

Consider:

y = −2x + 5

Identify:

m = −2 = −2/1

b = 5

Plot:

(0, 5)

Then move:

• 1 right
• 2 down

This gives:

(1, 3)

Repeat:

(2, 1)

The line falls from left to right.

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Graphing a Fractional Slope

Consider:

y = (3/4)x − 2

Identify:

m = 3/4

b = −2

Plot the y-intercept:

(0, −2)

Use the slope:

rise = 3

run = 4

Move:

• 4 right
• 3 up

This gives:

(4, 1)

Repeat the pattern to find more points.

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Moving in the Opposite Direction

A slope of:

3/4

can also be represented by:

−3/−4

So instead of moving:

4 right and 3 up

you can move:

4 left and 3 down.

Starting from:

(0, −2)

this produces:

(−4, −5)

Both methods remain on the same line.

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Graphing a Negative Fractional Slope

Consider:

y = −(2/3)x + 4

Plot:

(0, 4)

Since:

m = −2/3

you can move:

• 3 right
• 2 down

giving:

(3, 2)

Or move:

• 3 left
• 2 up

giving:

(−3, 6)

All three points lie on the same line.

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A Reliable Graphing Method

For an equation in slope-intercept form:

1. identify m
2. identify b
3. plot the y-intercept (0, b)
4. write m as rise/run
5. use the rise and run to locate another point
6. repeat if useful
7. check the points
8. draw a straight line through them
9. extend the line in both directions when appropriate

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Writing an Equation from Slope and Intercept

Suppose:

slope = 4

y-intercept = 7

Start with:

y = mx + b

Substitute:

m = 4

b = 7

Therefore:

y = 4x + 7

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Another Example

Suppose:

slope = −3

y-intercept = 5

Substitute into:

y = mx + b

The equation is:

y = −3x + 5

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Writing an Equation with a Fractional Slope

Suppose:

m = 2/3

b = −4

Then:

y = (2/3)x − 4

This equation completely describes the line.

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Determining an Equation from a Graph

To determine the equation of a line from a graph:

1. identify the y-intercept
2. choose another clear point on the line
3. calculate the slope
4. substitute m and b into y = mx + b
5. check the equation using another point

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

(0, 2)

and contains:

(3, 8)

The slope is:

m = (8 − 2)/(3 − 0)

m = 6/3

m = 2

The intercept is:

b = 2

Therefore:

y = 2x + 2

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When the y-Intercept Is Negative

Suppose a graph crosses the y-axis at:

(0, −5)

and contains:

(2, 1)

Calculate slope:

m = (1 − (−5))/(2 − 0)

m = 6/2

m = 3

Therefore:

y = 3x − 5

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Finding the Equation from Two Points

Sometimes two points are known but the y-intercept is not immediately given.

Suppose the line passes through:

(2, 7)

and:

(5, 16)

First calculate slope:

m = (16 − 7)/(5 − 2)

m = 9/3

m = 3

Now use:

y = 3x + b

Substitute (2, 7):

7 = 3(2) + b

7 = 6 + b

b = 1

Therefore:

y = 3x + 1

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Checking the Equation

Use the second point:

(5, 16)

Substitute:

16 = 3(5) + 1

16 = 15 + 1

16 = 16

The equation is verified.

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Determining an Equation from a Table

Consider:

The change in y is:

+3

for each:

+1 in x

Therefore:

m = 3

When x = 0:

y = 4

Therefore:

b = 4

The equation is:

y = 3x + 4

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When x = 0 Is Not in the Table

Consider:

Calculate slope:

m = (15 − 9)/(4 − 2)

m = 6/2

m = 3

Now use:

y = 3x + b

Substitute:

(2, 9)

9 = 3(2) + b

9 = 6 + b

b = 3

Therefore:

y = 3x + 3

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Connecting Tables, Graphs, and Equations

Consider:

y = 2x + 5

Equation:

m = 2

b = 5

Table:

Graph:

• crosses the y-axis at (0, 5)
• rises 2 for every movement of 1 to the right

All three representations show the same relationship.

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Slope-Intercept Form and Direct Variation

Direct variation has the form:

y = kx

Slope-intercept form is:

y = mx + b

For direct variation:

b = 0

Therefore:

y = mx

and the graph passes through:

(0, 0)

For example:

y = 5x

has:

m = 5

b = 0

It is both linear and directly proportional.

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Comparing Two Lines

Consider:

A: y = 2x + 8

B: y = 5x + 2

Line A:

m = 2

b = 8

Line B:

m = 5

b = 2

Therefore:

• A has the greater y-intercept
• B has the greater slope
• A starts higher
• B increases more quickly

Slope-intercept form makes these comparisons particularly easy.

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Same Slope, Different Intercepts

Consider:

y = 3x + 2

and:

y = 3x − 6

Both have:

m = 3

Their y-intercepts are different.

Therefore, their graphs are parallel.

They increase at exactly the same rate but begin at different vertical positions.

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Same Intercept, Different Slopes

Consider:

y = 2x + 4

and:

y = −3x + 4

Both cross the y-axis at:

(0, 4)

But their slopes differ.

The first line increases.

The second line decreases.

Therefore, the lines move away from their shared intercept in different directions.

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Interpreting Slope in Context

Suppose:

C = 4d + 7

where:

C = taxi cost in dollars

d = distance in kilometres

The slope is:

4 dollars/km

This means the taxi cost increases by $4 for every kilometre travelled.

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Interpreting the Intercept in Context

For:

C = 4d + 7

the intercept is:

7

At:

d = 0

C = 7

Therefore, the taxi charges a:

$7 starting fee

The intercept represents the initial value before any distance is travelled.

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Real-World Example: Hourly Pay

Suppose:

P = 20h + 50

where:

P = total pay

h = hours worked

Slope:

m = 20 dollars/hour

Interpretation:

The worker earns an additional $20 for each hour worked.

Intercept:

b = 50 dollars

Interpretation:

The worker begins with a $50 fixed payment or bonus.

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Real-World Example: Water Tank

Suppose:

V = −15t + 300

where:

V = water volume in litres

t = time in minutes

Slope:

−15 L/min

This means the tank loses 15 L every minute.

Intercept:

300 L

This means the tank contains 300 L at:

t = 0

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Real-World Example: Temperature

Suppose:

T = −2t + 24

where:

T = temperature in °C

t = time in hours

Slope:

−2°C/hour

The temperature decreases by 2°C each hour.

Intercept:

24°C

The initial temperature is 24°C.

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Real-World Example: Savings

Suppose:

S = 35w + 120

where:

S = savings in dollars

w = number of weeks

Slope:

$35/week

The person saves an additional $35 each week.

Intercept:

$120

The person already had $120 at week zero.

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Why Units Matter

Suppose an equation has:

m = 5

Without context, this tells us only the mathematical rate.

Depending on the variables, it might mean:

• $5 per item
• 5 km per hour
• 5 L per minute
• 5°C per hour
• 5 people per year

The units of slope come from:

y-units/x-units

The intercept uses the same units as y.

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When an Intercept May Not Be Realistic

Suppose:

P = 25a − 100

models profit P based on the number of items a sold.

Mathematically:

when a = 0:

P = −100

The intercept represents:

−$100

This could represent a fixed initial cost.

Sometimes the intercept is mathematically meaningful even when it is negative.

In other situations, x = 0 might fall outside the realistic domain, so the intercept may have little practical meaning.

Always interpret the equation within its context.

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Rearranging into Slope-Intercept Form

Not every linear equation is initially written as:

y = mx + b

Consider:

2x + y = 7

Solve for y.

Subtract 2x:

y = −2x + 7

Now we can identify:

m = −2

b = 7

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Another Rearranging Example

Consider:

3x + 2y = 10

Subtract 3x:

2y = −3x + 10

Divide everything by 2:

y = −(3/2)x + 5

Therefore:

m = −3/2

b = 5

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Watch the Signs When Rearranging

Consider:

4x − y = 6

Subtract 4x:

−y = −4x + 6

Now divide everything by −1:

y = 4x − 6

Therefore:

m = 4

b = −6

A common mistake is forgetting that dividing by −1 changes the signs of every term.

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Determining Whether an Equation Is in Slope-Intercept Form

Consider:

y = 5x − 3

Yes. It is already in:

y = mx + b

Consider:

2x + y = 8

No. It must first be rearranged.

Consider:

x = 4

No. This is a vertical line and cannot be expressed in slope-intercept form.

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Finding an Unknown Value Using the Equation

Suppose:

y = 3x + 7

Find y when x = 6.

Substitute:

y = 3(6) + 7

y = 18 + 7

y = 25

Therefore:

(6, 25)

lies on the line.

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Finding x from a Known y-Value

Suppose:

y = 4x + 5

and:

y = 29

Substitute:

29 = 4x + 5

Subtract 5:

24 = 4x

Divide by 4:

x = 6

Therefore:

(6, 29)

lies on the line.

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Verifying a Point

Does:

(4, 13)

lie on:

y = 3x + 1?

Substitute:

13 = 3(4) + 1

13 = 12 + 1

13 = 13

Yes.

Therefore, the point lies on the line.

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Detecting an Incorrect Equation

Suppose a graph appears to have:

slope = 2

y-intercept = −3

A student writes:

y = −3x + 2

The student has switched the slope and intercept.

The correct equation is:

y = 2x − 3

Always remember:

y = mx + b

m = slope

b = y-intercept

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From Words to Slope-Intercept Form

Suppose:

"A gym charges a $30 registration fee and $12 per visit."

Let:

C = total cost

v = number of visits

Starting value:

30

Rate:

12 per visit

Therefore:

C = 12v + 30

This has exactly the same structure as:

y = mx + b

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From Slope-Intercept Form to Words

Consider:

D = 75t + 20

Suppose D is distance in kilometres and t is time in hours.

A possible interpretation is:

"An object begins 20 km from the reference point and moves away at 75 km/h."

The slope gives the rate.

The intercept gives the initial position.

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Comparing Representations

Slope and intercept can be identified from different representations.

Being able to move between these representations is one of the main advantages of understanding slope-intercept form.

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A Reliable Strategy for Graphing y = mx + b

1. Make sure the equation is written as y = mx + b.
2. Identify m.
3. Identify b.
4. Plot the y-intercept (0, b).
5. Write the slope as a fraction.
6. Use rise/run to locate another point.
7. Repeat the slope pattern if needed.
8. Check the coordinates.
9. Draw a straight line through the points.
10. Extend the line appropriately.

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A Reliable Strategy for Finding an Equation from a Graph

1. Locate the y-intercept.
2. Record b.
3. Choose another clear point on the line.
4. Calculate rise and run.
5. Find m.
6. Write y = mx + b.
7. Substitute m and b.
8. Simplify the equation.
9. Test another point from the graph.
10. Check that the slope direction matches the graph.

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Common Mistakes

Switching m and b

In:

y = 4x + 7

m = 4

b = 7

not the other way around.

Giving b instead of the y-intercept point

If b = 5, the y-intercept is:

(0, 5)

Missing an implied slope

In:

y = x + 3

m = 1

In:

y = −x + 3

m = −1

Forgetting that a whole-number slope has denominator 1

m = 4 means:

4/1

Moving the wrong direction for a negative slope

A negative slope can be shown by moving right and down.

Using the x-intercept instead of the y-intercept

The y-intercept occurs where:

x = 0

Reading slope only from visual steepness

Use rise/run or coordinates whenever possible.

Forgetting to solve for y first

In:

2x + y = 8

the slope is not 2.

Rearrange:

y = −2x + 8

so the slope is −2.

Forgetting to change all signs when dividing by a negative

Be especially careful when isolating y.

Trying to write a vertical line in slope-intercept form

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

Ignoring context

Slope and intercept should be interpreted using appropriate quantities and units.

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Key Terms

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

Slope — rate of change represented by m.

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

Rate of change — change in y compared with change in x.

Rise — vertical change between points.

Run — horizontal change between points.

Coefficient — number multiplying a variable.

Constant term — term without a variable; b in slope-intercept form.

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

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

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line.

Linear equation — equation whose graph forms a straight line.

Initial value — starting value of the dependent variable, often represented by the y-intercept.

Parallel lines — non-intersecting lines with equal slopes.

Direct variation — proportional relationship of the form y = kx, equivalent to slope-intercept form with b = 0.

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Key Takeaways

• Slope-intercept form is y = mx + b.
• The value m represents the slope.
• The value b represents the y-intercept.
• The y-intercept occurs at (0, b).
• Slope represents the rate of change.
• Positive slopes rise from left to right.
• Negative slopes fall from left to right.
• Zero slope produces a horizontal line.
• Vertical lines have undefined slope and cannot be written in slope-intercept form.
• Whole-number slopes can be written over 1 to show rise/run.
• Fractional slopes directly show a rise/run pattern.
• To graph from slope-intercept form, plot the y-intercept first and then use the slope.
• A line can be written in slope-intercept form when its slope and y-intercept are known.
• If two points are known, calculate the slope and then determine b.
• Tables can be used to determine both slope and intercept.
• Equations may need to be rearranged to isolate y before m and b can be identified.
• Lines with the same slope and different intercepts are parallel.
• Direct variation is the special case in which b = 0.
• In real situations, slope usually represents a rate such as dollars/hour, km/h, or L/min.
• The y-intercept often represents a starting amount, initial condition, or fixed charge.
• The intercept should only be interpreted within the realistic domain of the situation.
• Tables, graphs, equations, and verbal descriptions can all communicate the same slope and intercept.

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Check Your Understanding

1. Identify the slope and y-intercept of:

y = 5x + 8

2. Identify the slope and y-intercept of:

y = −3x − 4

3. Identify m and b in:

y = (2/5)x + 7

Then describe how you could use them to graph the line.

4. Determine the slope and y-intercept of:

y = −x + 6

5. Write an equation in slope-intercept form with:

m = 4

and:

b = −7

6. Write an equation for a line with slope −3/2 and y-intercept (0, 5).
7. Describe how you would graph:

y = 2x − 3

using only its slope and y-intercept.

8. A line crosses the y-axis at (0, 4) and also passes through (3, 13). Calculate its slope and write its equation in slope-intercept form.
9. Determine the equation of the line passing through (2, 5) and (6, 17).
10. Write this equation in slope-intercept form and identify m and b:

2x + y = 9

11. Write this equation in slope-intercept form:

3x + 2y = 12

Then identify its slope and y-intercept.

12. A taxi company's cost is represented by:

C = 2.75d + 6

where d is distance in kilometres. Explain the meaning of both 2.75 and 6, including appropriate units.

13. A tank's water volume is represented by:

V = −18t + 450

where V is measured in litres and t in minutes. Identify and interpret the slope and intercept. What does the negative slope tell you?

14. Two lines are represented by:

A: y = 4x + 9

B: y = 4x − 3

Compare their slopes and y-intercepts. Explain what their graphs would look like relative to each other.

15. A student looks at a graph and determines that its y-intercept is −2. The graph also contains the points (3, 4) and (6, 10). Determine the slope, write the equation in slope-intercept form, verify that all the information is consistent, and explain how you would graph the line using only m and b.