Linear Relationship Examples in Math

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

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 relationship between two variables where the rate of change is constant, producing a straight line when graphed. Expressed as $y = mx + b$ where $m$ is the slope.

Add the same amount each step. Like paying \$10/month—increase is constant.

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: A linear relationship adds the same amount each step, drawing a straight line $y=mx+b$ .

Common stuck point: The procedure for linear relationship is the easy part; the trap is assuming any increasing pattern is linear. Asking "Does each equal step in $x$ add the same fixed amount to $y$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does each equal step in $x$ add the same fixed amount to $y$ ?

Worked Examples

Example 1

easy

A taxi charges a $3 base fee plus $2 per mile. Write the equation for total cost

C

in terms of miles

m

. Identify slope and y-intercept.

Answer

C = 2m + 3

; slope = 2, y-intercept = 3

First step

Base fee (y-intercept):

b = 3

Full solution

2
Cost per mile (slope): $m_{\text{rate}} = 2$ .
3
Equation: $C = 2m + 3$ .
4
This is in the form $y = mx + b$ with slope 2 and y-intercept 3.

A linear relationship

y = mx + b

has constant slope

m

(rate of change) and y-intercept

b

(starting value).

Example 2

medium

Two points on a line are

(1, 5)

and

(3, 11)

. Find the equation of the line in

y = mx + b

form.

Example 3

easy

A phone plan costs $25/month plus $0.10 per minute. Write the linear cost equation for

m

minutes used in a month.

Example 4

medium

A gym charges a \$50 sign-up fee plus \$30 per month. After how many months will the total cost be \$260?

Example 5

medium

A car's gas tank holds

15

gallons. The car uses

3

gallons per hour of driving. Write a linear equation for gallons left

G

after

t

hours.

Example 6

hard

A line has slope

\tfrac{2}{3}

and passes through

(-3, -1)

. Find the equation and the

x

-intercept.

Practice Problems

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

Example 1

easy

For the equation

y = 4x - 1

, find

y

when

x = 3

and

x = 0

Example 2

medium

Points

(0, -2)

and

(4, 6)

lie on a line. Find the equation in

y = mx + b

form.

Example 3

easy

y = 2x + 3

a linear relationship?

Example 4

easy

y = 4x - 1

, what is the slope?

Example 5

easy

y = 4x - 1

, what is the

y

-intercept?

Example 6

easy

A table adds

5

y

each time

x

rises by

1

. Is it linear?

Example 7

easy

Find

y

when

x = 2

for

y = 3x + 1

Example 8

easy

Does the line

y = 2x + 3

pass through the origin?

Example 9

easy

What is the slope of a line through

(0,2)

and

(1,5)

Example 10

easy

y = 7

(a horizontal line) linear?

Example 11

medium

A table:

(x,y) = (0,1),(1,4),(2,7),(3,10)

. Is it linear? Give the equation.

Example 12

medium

Find the equation of the line through

(1,5)

and

(3,11)

Example 13

medium

A taxi charges $3 base plus $2 per mile. Write the linear equation and find the cost for

5

miles.

Example 14

medium

Is the table

(x,y)=(1,2),(2,4),(3,8)

linear?

Example 15

medium

A line has slope

2

and passes through

(4, 10)

. Find

b

Example 16

medium

For

y = -2x + 6

, where does the line cross the

x

-axis?

Example 17

medium

Two siblings save money: A starts at \$10 and adds \$5/week; B starts at \$0 and adds \$5/week. Are both linear? Which is proportional?

Example 18

challenge

A line passes through

(2,7)

and

(5,16)

. Find

y

when

x = 8

without graphing.

Example 19

challenge

Show that a relationship with constant first differences in

y

(for equal

x

steps) must satisfy

y = mx + b

Example 20

challenge

A line and a proportional relation both pass through

(4, 12)

. The line also passes through

(0, 4)

. Find both equations.

Example 21

medium

A candle burns down linearly:

20

cm tall at

0

hours,

14

cm at

3

hours. Find the burn rate and the equation.

Example 22

medium

Find

x

when

y = 17

for the line

y = 4x + 1

Example 23

easy

For

y = 5x - 7

, what is the slope?

Example 24

easy

For

y = -3x + 8

, what is the

y

-intercept?

Example 25

easy

Find

y

when

x = 5

for

y = -2x + 11

Example 26

easy

A pool is filled at a constant rate. After

2

hours it has

40

gallons; after

5

hours it has

100

gallons. What is the rate (gallons per hour)?

Example 27

medium

Two points on a line are

(2, 9)

and

(6, 21)

. Find the slope.

Example 28

medium

A line passes through

(0, -4)

and

(3, 5)

. Write its equation in

y = mx + b

form.

Example 29

medium

A line has slope

-2

and passes through

(3, 4)

. Find its

y

-intercept

b

Example 30

medium

For the equation

2x + 3y = 12

, find the slope and

y

-intercept.

Example 31

medium

A linear table has

(x,y) = (0, 7), (1, 4), (2, 1), (3, -2)

. Write the equation.

Example 32

medium

Find

x

when

y = -5

for the line

y = 2x - 11

Example 33

medium

A line passes through

(-2, 1)

with slope

\tfrac{1}{2}

. Find its equation in

y = mx + b

form.

Example 34

medium

A line has

x

-intercept

4

and

y

-intercept

-6

. Find the equation in

y = mx + b

form.

Example 35

medium

A table shows hours

h

and pay

P

(2, 30), (5, 75), (8, 120)

. Write the linear pay equation.

Example 36

hard

A line passes through

(1, 4)

and is parallel to

y = -3x + 7

. Find its equation.

Example 37

hard

A submarine descends linearly. At

t = 0

min it is at depth

40

m. At

t = 6

min it is at depth

130

m. Find the depth at

t = 10

min.

Example 38

hard

Lines

A

and

B

both pass through

(4, 10)

. Line

A

also passes through

(0, 2)

and is linear but not proportional. Line

B

is proportional. Find both equations.

Example 39

hard

A line passes through

(-2, 5)

and

(4, -7)

. Find the value of

y

when

x = 0

(the

y

-intercept).

Example 40

challenge

A linear function

f

satisfies

f(3) = 11

and

f(7) = 23

. Find

f(100)

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Related Concepts

Rate of Change Linear Functions Slope

Background Knowledge

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

rate of change