Linear equations in two variables and their graphs

Warm-up

Three quick questions from earlier lessons.

Discover

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

Yes
No

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 $-A/B$ .

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