Graphical solutions and intersection checks | Year 8

01 / LEARN

The idea, step by step

One point satisfies both rules

At an intersection, the same ordered pair belongs to both graphs. Read both coordinates: an input alone does not identify the complete common point.

A graph can suggest a solution but may only give an estimate. Substitute an exact candidate into both original rules to confirm it.

Recognise when a picture is approximate

If the crossing falls between grid lines, state the precision of your reading rather than claiming an exact integer. A short equality calculation can confirm a fraction.

Parallel distinct lines have no common point. Identical graphs have every point in common, not just a single crossing.

A common trap

Giving only x = 1 as a graph intersection.

Instead: Give an ordered pair unless the question asks only for an input.

02 / WORKED EXAMPLES

See the thinking

Example 1

A visible common point

Find the intersection of $y = 2 x + 1$ and $y = - x + 4$ .

The graph suggests x = 1 and y = 3.
At x = 1, the first rule gives 3 and the second also gives 3.
Both coordinates are therefore confirmed.

(1, 3)

Example 2

A horizontal partner

Find the intersection of $y = x + 2$ and $y = 4$ .

The second rule fixes y at 4.
Solve $4 = x + 2$ , so x = 2.
Check the first output: $2 + 2 = 4$ .

(2, 4)

Check your understanding

Does $(2, 5)$ lie on both $y = 2 x + 1$ and $y = x + 3$ ?

Show answer

Yes.

At x = 2 both outputs equal 5.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Verify that $(1, 4)$ lies on $y = 3 x + 1$ and $y = x + 3$ .
2 marks
Find the intersection of $y = x + 1$ and $y = 3$ .
2 marks
Find the intersection of $y = 2 x$ and $y = x + 2$ .
2 marks
Find the intersection of $y = - x + 5$ and $y = x + 1$ .
2 marks
A crossing of $y = 2 x + 1$ and $y = x + 3$ is read as $(2, 6)$ . Use checks to assess this.
3 marks
Can $y = 2 x + 1$ and $y = 2 x - 3$ intersect? Explain.
3 marks

Read the worked solutions

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

Verify that $(1, 4)$ lies on $y = 3 x + 1$ and $y = x + 3$ .
2 marks
First output: $3 + 1 = 4$ .
Second output: $1 + 3 = 4$ .
It lies on both.
Find the intersection of $y = x + 1$ and $y = 3$ .
2 marks
Solve $x + 1 = 3$ .
Then x = 2 and y = 3.
$(2, 3)$ .
Find the intersection of $y = 2 x$ and $y = x + 2$ .
2 marks
Set $2 x = x + 2$ , giving x = 2.
The shared output is 4.
$(2, 4)$ .
Find the intersection of $y = - x + 5$ and $y = x + 1$ .
2 marks
Set $- x + 5 = x + 1$ , giving $2 x = 4$ .
Substitute x = 2 to get y = 3.
$(2, 3)$ .
A crossing of $y = 2 x + 1$ and $y = x + 3$ is read as $(2, 6)$ . Use checks to assess this.
3 marks
Both rules give 5 at x = 2.
The proposed x works, but the y-reading is one unit too high.
The correct point is $(2, 5)$ .
Can $y = 2 x + 1$ and $y = 2 x - 3$ intersect? Explain.
3 marks
At every input the first output is 4 larger.
The graphs are parallel distinct lines.
No.

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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 ↗

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