Open MathNSW GitHub ↗

Open Math›Year 9›Linear relationships›Graphing linear equations

Graphing linear equations

Graph linear equations by finding intercepts and using gradient-intercept form. Determine the equation of a line from a graph or two points.

Worked examples

Finding intercepts to graph a line

Straightforward

Problem

Find the

x

-intercept and

y

-intercept of

y = 3x - 9

, and use them to sketch the line.

1

Find the

y

-intercept by setting

x = 0

.

y = 3(0) - 9 = -9

. The

y

-intercept is

(0, -9)

.

2

Find the

x

-intercept by setting

y = 0

and solving for

x

.

0 = 3x - 9 \Rightarrow 3x = 9 \Rightarrow x = 3

. The

x

-intercept is

(3, 0)

.

3

Plot both intercepts and draw a straight line through them.

Plot

(0, -9)

and

(3, 0)

. Draw a line through these two points.

Answer

x

-intercept:

(3, 0)

;

y

-intercept:

(0, -9)

Rearranging to gradient-intercept form

Moderate

Problem

Rearrange

3x - 4y = 20

into gradient-intercept form and state the gradient and

y

-intercept.

1

Isolate the

y

-term by subtracting

3x

from both sides.

-4y = -3x + 20

2

Divide every term by

-4

.

y = \dfrac{-3x + 20}{-4} = \dfrac{3}{4}x - 5

3

Read off the gradient and

y

-intercept from

y = mx + b

.

Gradient

m = \dfrac{3}{4}

;

y

-intercept

b = -5

, i.e. the point

(0, -5)

.

Answer

y = \dfrac{3}{4}x - 5

; gradient

= \dfrac{3}{4}

,

y

-intercept

= -5

Finding the equation of a line through two points

Challenging

Problem

Find the equation of the line passing through

(-1, 5)

and

(3, -3)

.

1

Calculate the gradient using

m = \dfrac{y_2 - y_1}{x_2 - x_1}

.

m = \dfrac{-3 - 5}{3 - (-1)} = \dfrac{-8}{4} = -2

2

Write the equation as

y = -2x + b

and substitute one point to find

b

.

Using

(-1, 5)

:

5 = -2(-1) + b \Rightarrow 5 = 2 + b \Rightarrow b = 3

3

Write the final equation and verify with the second point.

Equation:

y = -2x + 3

. Check with

(3, -3)

:

y = -2(3) + 3 = -3

✓

Answer

y = -2x + 3

Practise

Q1·Straightforward

Find the

x

-intercept of the line

y = 2x - 6

.

Explanation

Set

y = 0

:

0 = 2x - 6

, so

2x = 6

, giving

x = 3

. The

x

-intercept is

(3, 0)

.

Q2·Straightforward

Find the

y

-intercept of the line

y = -3x + 5

.

Explanation

Set

x = 0

:

y = -3(0) + 5 = 5

. The

y

-intercept is

(0, 5)

.

Q3·Straightforward

Find the

x

-intercept of the line

y = 4x + 8

.

Explanation

Set

y = 0

:

0 = 4x + 8

, so

4x = -8

, giving

x = -2

. The

x

-intercept is

(-2, 0)

.

Q4·Straightforward

Find the

y

-intercept of the line

3x + y = 7

.

Explanation

Set

x = 0

:

3(0) + y = 7

, so

y = 7

. The

y

-intercept is

(0, 7)

.

Q5·Moderate

Rearrange

2x + 3y = 12

into gradient-intercept form

y = mx + b

. What is the gradient

m

?

/

Explanation

Subtract

2x

:

3y = -2x + 12

. Divide by

3

:

y = -\dfrac{2}{3}x + 4

. The gradient is

-\dfrac{2}{3}

.

Q6·Moderate

Rearrange

5x - y = 3

into gradient-intercept form. What is the

y

-intercept?

Explanation

Subtract

5x

:

-y = -5x + 3

. Multiply by

-1

:

y = 5x - 3

. The

y

-intercept is

-3

.

Q7·Moderate

Rearrange

4x - 6y = 18

into gradient-intercept form. What is the gradient?

/

Explanation

Subtract

4x

:

-6y = -4x + 18

. Divide by

-6

:

y = \dfrac{2}{3}x - 3

. The gradient is

\dfrac{2}{3}

.

Q8·Moderate

Rearrange

x - 2y = 8

into gradient-intercept form. What is the

y

-intercept?

Explanation

Subtract

x

:

-2y = -x + 8

. Divide by

-2

:

y = \dfrac{1}{2}x - 4

. The

y

-intercept is

-4

.

Q9·Challenging

A line passes through the points

(1, 3)

and

(4, 9)

. What is the equation of this line?

$y = 2x + 1$$y = 3x$$y = 2x - 1$$y = 3x - 3$

Explanation

Gradient:

m = \dfrac{9 - 3}{4 - 1} = \dfrac{6}{3} = 2

. Using

y = 2x + b

with

(1, 3)

:

3 = 2(1) + b

, so

b = 1

. Equation:

y = 2x + 1

.

Q10·Challenging

A line has

x

-intercept

4

and

y

-intercept

-2

. What is the equation of this line?

$y = \dfrac{1}{2}x - 2$$y = -\dfrac{1}{2}x + 2$$y = 2x - 2$$y = \dfrac{1}{2}x + 2$

Explanation

Gradient:

m = \dfrac{0 - (-2)}{4 - 0} = \dfrac{2}{4} = \dfrac{1}{2}

. The

y

-intercept is

-2

, so the equation is

y = \dfrac{1}{2}x - 2

.

Q11·Challenging

A line passes through

(-2, 7)

and

(3, -3)

. What is the equation of this line?

$y = -2x + 3$$y = 2x + 3$$y = -2x - 3$$y = -\dfrac{1}{2}x + 5$

Explanation

Gradient:

m = \dfrac{-3 - 7}{3 - (-2)} = \dfrac{-10}{5} = -2

. Using

y = -2x + b

with

(3, -3)

:

-3 = -2(3) + b

, so

b = 3

. Equation:

y = -2x + 3

.

Q12·Challenging

A straight line graph passes through

(0, -4)

and has gradient

\dfrac{3}{2}

. Which equation matches this line?

$y = \dfrac{3}{2}x - 4$$y = \dfrac{2}{3}x - 4$$y = \dfrac{3}{2}x + 4$$y = -\dfrac{3}{2}x - 4$

Explanation

The

y

-intercept is

(0, -4)

so

b = -4

. With gradient

m = \dfrac{3}{2}

, the equation is

y = \dfrac{3}{2}x - 4

.

Built by Robert Webster at 1up ITFree forever · No accounts · Open source