Proportional relationships: y = kx

3 pounds of apples cost 6 dollars. At the same price, what do 5 pounds cost?

Proportional means $y = kx$ .

In a proportional relationship

y = kx

, where

k = \tfrac{y}{x}

is the constant of proportionality (the unit rate). Its graph is a line through the origin, and

k

is the slope.

A printer prints 45 pages in 3 minutes.

k = 15

pages per minute

45 ÷ 3.

y = 15x

Equation.

120 pages in 8 minutes

x = 8.

Your turn: $y$ is proportional to $x$ , and $y = 28$ when $x = 4$ . Find $k$ .

7

28 ÷ 4.

$y$ is proportional to $x$ . When $x = 2$ , $y = 10$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 3$ , $y = 15$ . Find the constant of proportionality $k$ in $y = kx$ .

Which equation represents a proportional relationship?
1. $y = 2x$
2. $y = 2x + 9$
3. $y = x + 2$

A train travels 21 miles in 3 minutes at a constant speed. How far does it travel in 12 minutes?
A train travels 30 miles in 6 minutes at a constant speed. How far does it travel in 10 minutes?

Three questions to check you have got it.

$y$ is proportional to $x$ . When $x = 2$ , $y = 10$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 3$ , $y = 15$ . Find the constant of proportionality $k$ in $y = kx$ .
$y$ is proportional to $x$ . When $x = 4$ , $y = 28$ . Find the constant of proportionality $k$ in $y = kx$ .

Proportional relationships

$y = kx$ , $k = y \div x$ ; the graph passes through $(0, 0)$ .

Watch out: scale by multiplying, not by adding.

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