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1. Proportional relationships: y = kx

Warm-up

Three quick questions from earlier lessons.

Discover

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

10 dollars
8 dollars

Key idea

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.

Worked example

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, guided

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

7

28 ÷ 4.

Practice ★

$y$ 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. $y = x + 2$

Watch out!

Practice ★★

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?

Exit ticket

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 $y = kx$ .

Summary card

Proportional relationships

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

Watch out: scale by multiplying, not by adding.

2. Linear relationships: slope-intercept form y = mx + b

Warm-up

Three quick questions from earlier lessons.

Discover

A taxi charges 3 dollars to start and 2 dollars per mile. Is the cost proportional to the distance?

No: it is linear, but starts at 3
Yes

Key idea

Every non-vertical line is $y = mx + b$ .

y = mx + b

m

is the slope (the rate of change) and

b

is the $y$ -intercept (the value at

x = 0

). When

b = 0

the relationship is proportional. Horizontal lines are

y = b

; vertical lines are

x = a

Worked example

Through $(0, 4)$ and $(3, 10)$ .

m = 2

(10 − 4) ÷ 3.

b = 4

Value at x = 0.

y = 2x + 4

Equation.

Your turn, guided

Your turn: the $y$ -intercept of $y = 7x - 5$ .

-5

Practice ★

A gym charges 55 dollars to join and 19 dollars per month. Write the total cost $y$ after $x$ months.
A gym charges 77 dollars to join and 16 dollars per month. Write the total cost $y$ after $x$ months.

Which is the equation of the horizontal line through $(5, 1)$ ?
1. $y = 1$
2. $x = 5$
3. $y = 5$

Watch out!

Practice ★★

A line passes through $(0, 2)$ and $(1, 8)$ . Write its equation in the form $y = mx + b$ .
A line passes through $(0, 2)$ and $(2, 12)$ . Write its equation in the form $y = mx + b$ .

Boss problem. A candle is 21 cm tall when lit and burns 3 cm per hour. Let $y$ be its height after $x$ hours.

Exit ticket

Three questions to check you have got it.

A gym charges 55 dollars to join and 19 dollars per month. Write the total cost $y$ after $x$ months.
A gym charges 77 dollars to join and 16 dollars per month. Write the total cost $y$ after $x$ months.
A gym charges 25 dollars to join and 17 dollars per month. Write the total cost $y$ after $x$ months.

Summary card

Slope-intercept form

$y = mx + b$ : rate $m$ , start $b$ .

Watch out: the rate multiplies $x$ ; the starting amount stands alone.

3. Linear equations in two variables and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

Is $(2, 4)$ a solution of $3x + y = 10$ ?

Key idea

A linear equation in two variables is a line.

The solutions of

Ax + By = C

are the points of a line. Set

y = 0

for the $x$ -intercept and

x = 0

for the $y$ -intercept. Solving for

y

gives

y = mx + b

, with slope

m = -\tfrac{A}{B}

Worked example

$2x + 5y = 20$ .

x

-intercept

10

y = 0.

y

-intercept

4

x = 0.

Slope

-\tfrac{2}{5}

y = −(2/5)x + 4.

Your turn, guided

Your turn: the $y$ -intercept of $3x + 4y = 24$ .

6

x = 0: 4y = 24.

Practice ★

The point $(2, y)$ is a solution of $5x + y = 15$ . Find $y$ .
The point $(1, y)$ is a solution of $5x + y = 17$ . Find $y$ .

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).

Watch out!

Practice ★★

What is the slope of the line $8x + 9y = 17$ ?
What is the slope of the line $1x + 4y = 4$ ?

Exit ticket

Three questions to check you have got it.

Find the $x$ -intercept of $3x + 6y = 54$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $3x + 1y = 3$ (the value of $x$ where the line crosses the $x$ -axis).
Find the $x$ -intercept of $4x + 1y = 12$ (the value of $x$ where the line crosses the $x$ -axis).

Summary card

Lines from equations

Intercepts: set the other variable to 0. Slope: solve for $y$ .

Watch out: slope of $Ax + By = C$ is $-A/B$ .

Answers: Linear relationships

1. Proportional relationships: y = kx

$y = 2x$

$84$
$50$

2. Linear relationships: slope-intercept form y = mx + b

$19 x + 55$
$16 x + 77$

$y = 1$

$6 x + 2$
$5 x + 2$

$19 x + 55$
$16 x + 77$
$17 x + 25$

3. Linear equations in two variables and their graphs

$5$
$12$

$18$

$- \frac{8}{9}$
$- \frac{1}{4}$

$18$
$1$
$3$