Slope Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Slope.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A measure of the steepness and direction of a line, calculated as the ratio of vertical change (rise) to horizontal change (run) between any two points on the line. Slope is written as $m = \frac{y_2 - y_1}{x_2 - x_1}$ and indicates how much $y$ changes for each unit increase in $x$ .

How much the line goes up for every step to the right. Steeper = bigger slope.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Slope is the steady rate a straight line climbs or falls.

Common stuck point: The procedure for slope is the easy part; the trap is computing $\frac{\Delta x}{\Delta y}$ instead of $\frac{\Delta y}{\Delta x}$ . Asking "Does the output change by the same amount for each equal step in the input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the output change by the same amount for each equal step in the input?

Worked Examples

Example 1

easy

Find the slope of the line passing through points

(2, 3)

and

(6, 11)

Find the slope of the line through A and B.

Answer

m = 2

First step

Use the slope formula:

m = \frac{y_2 - y_1}{x_2 - x_1}

Full solution

2
Substitute: $m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2$ .
3
The slope is 2, meaning the line rises 2 units for every 1 unit to the right.

Slope measures steepness: the ratio of vertical change (rise) to horizontal change (run). A positive slope means the line goes upward from left to right.

Example 2

medium

A line passes through

(3, 7)

and

(3, -2)

. What is its slope?

A line through (3,7) and (3,−2) — what is its slope?

Example 3

medium

Find the slope of the line passing through

(2, 5)

and

(6, 13)

Find the slope through A(2,5) and B(6,13).

Example 4

easy

A ramp rises

3

ft over a

15

ft horizontal run. What is its slope?

Example 5

medium

A table shows

(1, 4), (2, 7), (3, 10), (4, 13)

. Is the relationship linear, and if so, what is the slope?

Four points from a table — is the pattern linear, and what is the slope?

Example 6

medium

A car's distance after

t

hours is

60t + 20

miles. What does the slope

60

represent?

Example 7

medium

A staircase has

7

steps; each step is

8

in tall and

11

in deep. What is the staircase's slope?

Example 8

hard

A line has

x

-intercept

4

and

y

-intercept

-6

. Find its slope.

Find the slope of the line with x-int 4 and y-int −6.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the slope of the line through

(1, 4)

and

(5, 12)

Find the slope through A(1,4) and B(5,12).

Example 2

hard

A line has slope

-\frac{3}{4}

and passes through

(0, 5)

. Find the

y

-value when

x = 8

Example 3

easy

Find the slope of the line through

(1, 2)

and

(4, 8)

Find the slope through A(1,2) and B(4,8).

Example 4

easy

Find the slope of the line through

(0, 5)

and

(2, 1)

Find the slope through A(0,5) and B(2,1).

Example 5

easy

Find the slope of the line through

(3, 7)

and

(5, 7)

Find the slope through A(3,7) and B(5,7).

Example 6

easy

Find the slope of the line through

(2, 3)

and

(2, 8)

What is the slope of the line through (2,3) and (2,8)?

Example 7

easy

Slope of

y = 3x + 2

Identify the slope of y = 3x + 2.

Example 8

easy

Slope of

y = -\frac{1}{2}x + 4

Identify the slope of y = −½x + 4.

Example 9

easy

Find the slope of the line

y = 5

Example 10

easy

A roof rises

4

feet over a horizontal distance of

10

feet. What is the slope?

Example 11

medium

Find the slope of

2x + 3y = 12

Find the slope of 2x + 3y = 12.

Example 12

medium

Are lines

y = 2x + 1

and

y = 2x - 5

parallel?

Example 13

medium

Are lines with slopes

3

and

-\frac{1}{3}

perpendicular?

Example 14

medium

A line has slope

\frac{2}{3}

and passes through

(3, 1)

. Find its equation.

Line with slope ⅔ through (3,1): find its equation.

Example 15

medium

Find the slope between

(-2, 5)

and

(3, -10)

Find the slope between (−2,5) and (3,−10).

Example 16

medium

A line has slope

-2

and

y

-intercept

7

. Write its equation.

Example 17

medium

A line passes through

(1, 4)

and

(3, 10)

. Where does it cross the

y

-axis?

Where does the line through (1,4) and (3,10) cross the y-axis?

Example 18

medium

Find the slope of the line perpendicular to

y = \frac{1}{4}x + 3

Example 19

medium

What's the slope of a wheelchair ramp that rises

2

ft over a

24

-ft run?

Example 20

challenge

Lines

y = 2x + 3

and

y = mx - 1

are perpendicular. Find

m

Example 21

challenge

A line passes through

(2, 5)

with slope

3

. What's its

x

-intercept?

Line through (2,5) with slope 3 — find the x-intercept.

Example 22

challenge

Three points

(1, 2)

(4, 8)

(k, 14)

are collinear. Find

k

A(1,2), B(4,8), C(k,14) are collinear. Find k.

Example 23

easy

Find the slope of the line through

(0, 0)

and

(4, 12)

Find the slope of the line through O(0,0) and A(4,12).

Example 24

easy

What is the slope of the line

y = -7x + 9

What is the slope of y = −7x + 9?

Example 25

easy

Find the slope of the line through

(-1, -4)

and

(2, 5)

Find the slope through A(−1,−4) and B(2,5).

Example 26

medium

Find the slope of

4x - 2y = 10

Find the slope of 4x − 2y = 10.

Example 27

medium

A line passes through

(2, -1)

and has slope

4

. Find

y

when

x = 5

Line through (2,−1) with slope 4 — find y when x = 5.

Example 28

medium

Are the lines

y = \frac{1}{2}x + 3

and

2x - 4y = 7

parallel?

Example 29

medium

Find the slope of the line perpendicular to one with slope

\frac{5}{3}

Example 30

medium

Find the slope between

(\tfrac{1}{2}, 3)

and

(\tfrac{5}{2}, 11)

Find the slope through (½,3) and (5/2,11).

Example 31

medium

(4, k)

and

(6, 13)

lie on a line with slope

5

, find

k

Slope = 5 through (6,13); find k so (4,k) is on the line.

Example 32

hard

Find the slope of the line containing the midpoint of

(0, 0)

and

(8, 4)

and the point

(10, 9)

Find slope of line from midpoint of OB to P(10,9).

Example 33

hard

A line through

(1, 2)

and

(5, k)

is perpendicular to a line of slope

2

. Find

k

Line through (1,2) and (5,k) perpendicular to slope 2 — find k.

Example 34

hard

A line passes through

(2, 3)

with slope

-\frac{2}{5}

. Find its

x

-intercept.

Example 35

hard

Lines

y = (2a - 1)x + 3

and

y = (a + 5)x - 1

are parallel. Find

a

Example 36

hard

A pool drains at a steady rate. After

2

minutes it holds

480

gallons; after

7

minutes it holds

355

gallons. What is the slope (gallons per minute)?

Pool draining: find the slope (gallons per minute).

Example 37

challenge

Points

A(0, 0)

B(6, 8)

, and

C(c, 0)

form a right triangle with the right angle at

B

. Find

c

Right triangle ABC with right angle at B(6,8) — find c.

Example 38

challenge

Points

(2, 5)

(4, 11)

, and

(t, 2t + 1)

are collinear. Find

t

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Related Concepts

Coordinate Plane Rates Linear Functions Parallel and Perpendicular

Background Knowledge

These ideas may be useful before you work through the harder examples.

coordinate planerates