Proportional Line Examples in Math

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

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 straight line that passes through the origin, representing a proportional relationship of the form $y = kx$ with constant ratio $k$ .

When $x = 0$ , $y = 0$ . The line passes through the origin—no head start.

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 proportional line is $y=kx$ : a straight line passing through $(0,0)$ with a constant ratio $y/x=k$ .

Common stuck point: The procedure for proportional line is the easy part; the trap is calling a line with a nonzero $y$ -intercept proportional. Asking "Does the line pass through $(0,0)$ with $y/x$ the same for every point?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the line pass through $(0,0)$ with $y/x$ the same for every point?

Worked Examples

Example 1

easy

A recipe uses 3 cups of flour for every 2 cups of sugar. Write the relationship as

y = kx

and find

k

Answer

y = \frac{3}{2}x

First step

Let

x

= cups of sugar,

y

= cups of flour.

Full solution

2
The ratio is $\frac{y}{x} = \frac{3}{2}$ , so $k = \frac{3}{2}$ .
3
The proportional relationship is $y = \frac{3}{2}x$ .

A proportional relationship passes through the origin with equation

y = kx

. The constant

k

is the ratio of

y

x

and is the slope of the line.

Example 2

medium

Does the table represent a proportional relationship?

x

: 2, 4, 6;

y

: 5, 10, 15.

Example 3

medium

A car travels at a constant

60

km/hr. Write

d

as a proportional function of

t

and find

d

when

t=2.5

hr.

Example 4

medium

y

is proportional to

x

, and

y=18

when

x=12

. Find

y

when

x=20

Example 5

medium

Decide whether

\{(2,5),(4,10),(6,15)\}

lies on a proportional line. If so, give its equation.

Example 6

hard

The cost

C

to print flyers is proportional to the number

n

printed.

200

flyers cost $30. Write

C(n)

and find the cost of

750

flyers.

Example 7

hard

A graph shows a line passing through

(0,0)

(4,10)

, and a third point

(x,25)

. Find

x

Example 8

hard

y

varies directly with

x

. When

x

increases by

40\%

, by what percent does

y

change?

Example 9

hard

Graph

y=3x

. Identify a third lattice point on the line besides

(0,0)

and

(1,3)

Example 10

challenge

Show that the set of points satisfying both

y=kx

and

y=mx

(with

k\ne m

) is exactly

\{(0,0)\}

Practice Problems

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

Example 1

easy

y = 4x

, find

y

when

x = 7

Example 2

medium

y = 3x + 2

a proportional relationship? Why or why not?

Example 3

easy

y=3x

a proportional relationship?

Example 4

easy

y=2x+1

proportional?

Example 5

easy

Does the line

y=5x

pass through the origin?

Example 6

easy

Find the constant of proportionality in

y=7x

Example 7

easy

y=4x

, find

y

when

x=3

Example 8

easy

A table shows

(1,2),(2,4),(3,6)

. Is it proportional?

Example 9

easy

Write the equation of a proportional line with constant 6.

Example 10

easy

Is the line through

(0,0)

and

(2,6)

proportional, and what is

k

Example 11

medium

A car travels 150 miles in 3 hours at constant speed. Write the proportional relationship between distance

d

and time

t

Example 12

medium

The graph of

y=kx

passes through

(4,10)

. Find

k

and then

y

x=6

Example 13

medium

Two lines:

y=3x

and

y=3x+2

. Which is proportional and why?

Example 14

medium

y

is proportional to

x

and

y=20

when

x=5

, find

y

when

x=8

Example 15

medium

A recipe uses 2 cups of flour per 3 cookies. Write the proportional model and find flour for 12 cookies.

Example 16

medium

Determine whether

xy=12

describes a proportional line.

Example 17

challenge

A proportional line passes through

(a, 12)

and

(3, 4)

. Find

a

Example 18

challenge

Show that if

(x_1,y_1)

and

(x_2,y_2)

lie on

y=kx

, then

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

(for nonzero

x

Example 19

challenge

A line is proportional and also passes through

(2,5)

. A student claims it also passes through

(4,9)

. Verify and correct.

Example 20

medium

y

is proportional to

x

and

y=15

when

x=3

, find

k

and write the equation.

Example 21

medium

Is the table

(1,3),(2,5),(3,7)

proportional?

Example 22

medium

A proportional line has

y=8

x=2

. Find

y

x=5

Example 23

easy

y=8x

, find

y

when

x=5

Example 24

easy

A proportional line passes through

(2,10)

. Find

k

Example 25

easy

y=-3x

a proportional relationship?

Example 26

easy

Is the table

x

1,2,3

;

y

3,7,10

proportional?

Example 27

medium

A proportional line passes through

(4,-6)

. Write its equation.

Example 28

medium

Three of the points lie on the same proportional line:

(2,8),(3,12),(5,21)

. Which one does NOT?

Example 29

medium

A proportional graph passes through

(6,9)

. What is

y

when

x=10

Example 30

medium

For the proportional line

y=4x

, find

x

when

y=22

Example 31

hard

On a proportional line, the point

(a,12)

and the point

(3,4)

both lie on the line. Find

a

Example 32

hard

Lines

y=2x

and

y=3x

both pass through the origin. At what

x>0

does the second line's

y

-value exceed the first by

5

Example 33

hard

A proportional line passes through

(a,b)

with

a\ne 0

. Write the line's equation in terms of

a

and

b

Example 34

hard

Two proportional relationships share constants

k_1=2

and

k_2=5

. If

y_1=y_2

for some

x_1,x_2

, find

x_1/x_2

← Back to Proportional Line Formula Explained Practice Mode

Related Concepts

Linear Functions Proportionality Direct Variation Constant of Proportionality

Background Knowledge

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

linear functionsproportionality