Horizontal/vertical lines | Year 8

01 / LEARN

Understand the idea.

Horizontal lines fix the height

A horizontal line has the same height at every point. For $y = 3$ , the second coordinate stays $3$ while the first may vary; $(- 2, 3)$ , $(0, 3)$ and $(4, 3)$ are examples.

The equation $y = 3$ describes an entire line, not just $(0, 3)$ . Changing the first coordinate moves along that line without changing its height. The horizontal axis itself has equation $y = 0$ .

Vertical lines fix horizontal position

A vertical line fixes horizontal position. On $x = - 2$ , every first coordinate is $- 2$ , while the second can vary. Points line up above and below one another, two units left of the vertical axis.

The vertical axis has equation $x = 0$ . A vertical line has many heights for its one fixed input, so it cannot be expressed as a rule assigning one output $y$ to each input $x$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A horizontal line

Give three points on $y = - 2$ .

Equal-scale Cartesian axes from negative 4 to 4 have one-unit grid spacing. A sage horizontal line shows y equals negative 2.

Worked solution

1
Keep the required second coordinate.
$y = - 2$
Every point on the line must have height negative two.
2
Choose three different first coordinates.
$x = - 3, x = 0, x = 4$
The equation places no restriction on the first coordinate.
3
Pair each input with that height.
$(- 3, - 2), (0, - 2), (4, - 2)$
All three points have the required second coordinate and lie on the same horizontal line.

Answer

For example $(- 3, - 2), (0, - 2), (4, - 2)$ .

EXAMPLE 02

A fixed first coordinate

A line contains $(3, - 1), (3, 0), (3, 4)$ . What is its equation?

Worked solution

1
Compare the first coordinates.
$x = 3 for all three points$
Each listed point has the same horizontal position.
2
Compare the second coordinates.
$y = - 1, 0, 4$
The heights vary, so the line extends vertically.
3
Write the fixed-coordinate equation.
$x = 3$
These distinct points determine the vertical line three units right of the vertical axis.

Answer

$x = 3$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Give the equation of the x-axis.

Show answer & reasoning

$y = 0$ .

Every point on the horizontal axis has height zero, such as $(- 2, 0)$ and $(3, 0)$ . Its equation is therefore $y = 0$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Give two points on $y = 4$ .
2 marks
Give two points on $x = - 1$ .
2 marks
Which line is vertical: $x = 5$ or $y = 5$ ?
2 marks
A horizontal line passes through $(4, - 3)$ . Give its equation.
2 marks
A vertical line passes through $(- 4, 2)$ . Give its equation.
2 marks
Where do $x = 2$ and $y = - 3$ intersect?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Give two points on $y = 4$ .
2 marks
Keep $y$ equal to 4.
Choose different $x$ -values.
For example $(0, 4), (2, 4)$ .
Give two points on $x = - 1$ .
2 marks
Keep $x$ equal to -1.
Choose different $y$ -values.
For example $(- 1, 0), (- 1, 3)$ .
Which line is vertical: $x = 5$ or $y = 5$ ?
2 marks
It fixes horizontal position.
Its $y$ -values vary.
$x = 5$ .
A horizontal line passes through $(4, - 3)$ . Give its equation.
2 marks
Use the point's constant $y$ -coordinate.
The $x$ -coordinate may vary.
$y = - 3$ .
A vertical line passes through $(- 4, 2)$ . Give its equation.
2 marks
Use the point's constant $x$ -coordinate.
The $y$ -coordinate may vary.
$x = - 4$ .
Where do $x = 2$ and $y = - 3$ intersect?
2 marks
The point must satisfy both equations.
Combine $x = 2$ and $y = - 3$ .
$(2, - 3)$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Multi-step equations; brackets and variables on both sides; equations with rational solutions; substitution checks; one-variable inequalities and number-line solutions; tables and straight-line graphs; horizontal/vertical lines; how coefficients change a graph; graphical solutions and intersection checks; real-world linear models with digital exploration.

Official curriculum source ↗

In this lesson

Understand the idea.

Horizontal lines fix the height

Vertical lines fix horizontal position

See the method in action.

A horizontal line

Keep the required second coordinate.

Choose three different first coordinates.

Pair each input with that height.

A fixed first coordinate

Compare the first coordinates.

Compare the second coordinates.

Write the fixed-coordinate equation.

Try it yourself.

Check your understanding

Keep practising