Simultaneous equations by graphs, substitution and elimination | Year 10

01 / LEARN

The idea, step by step

A pair must satisfy both equations

A simultaneous solution is one ordered pair that makes both equations true. On a coordinate graph, it is the intersection of the two lines. A graph gives a useful estimate; substitution into both equations checks the coordinates.

Choose a method that fits

Substitution is efficient when one equation already isolates a variable. Elimination adds or subtracts equations after making one pair of coefficients equal or opposite. Always find both variables, write the answer as a pair, and check both original equations.

A common trap

Finding a point on one line and assuming that it solves the pair.

Instead: A simultaneous solution lies on both lines. Substitute the pair into both original equations.

02 / WORKED EXAMPLES

See the thinking

Example 1

Read and check an intersection

Solve $y = 2 x + 1$ and $y = 7 - x$ .

At an intersection the two values of $y$ match: $2 x + 1 = 7 - x$ .
$3 x = 6$ , so $x = 2$ and $y = 5$ .
The graph agrees; $5 = 2 (2) + 1$ and $5 = 7 - 2$ .

(x, y) = (2, 5)

Example 2

Eliminate a variable

Solve $2 x + 3 y = 13$ and $3 x - 2 y = 0$ .

Multiply the first by $2$ : $4 x + 6 y = 26$ .
Multiply the second by $3$ : $9 x - 6 y = 0$ .
Add to get $13 x = 26$ , hence $x = 2$ .
Use $6 - 2 y = 0$ to get $y = 3$ .

(2, 3)

Check your understanding

Solve $y = x + 2$ and $x + y = 8$ .

Show answer

$(3, 5)$

Substitute $y = x + 2$ : $2 x + 2 = 8$ , so $x = 3, y = 5$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Solve $x + y = 9$ and $x - y = 3$ .
2 marks
Solve $y = 3 x - 1$ and $y = x + 5$ .
2 marks
Solve $2 x + y = 11$ and $x + 2 y = 10$ .
2 marks
Solve $3 x - 2 y = 4$ and $x + y = 8$ .
2 marks
On the supplied grid, draw $y = x + 1$ and $y = 5 - x$ , then state their intersection.
3 marks
Three adult tickets and two child tickets cost 46 dollars. One adult ticket and two child tickets cost 26 dollars. Find both prices.
3 marks

Read the worked solutions

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

Solve $x + y = 9$ and $x - y = 3$ .
2 marks
Add the equations to get $2 x = 12$ ; then $y = 3$ .
$(6, 3)$
Solve $y = 3 x - 1$ and $y = x + 5$ .
2 marks
$3 x - 1 = x + 5$ gives $x = 3$ , then $y = 8$ .
$(3, 8)$
Solve $2 x + y = 11$ and $x + 2 y = 10$ .
2 marks
Eliminate $y$ : $4 x + 2 y = 22$ minus $x + 2 y = 10$ gives $3 x = 12$ .
$(4, 3)$
Solve $3 x - 2 y = 4$ and $x + y = 8$ .
2 marks
Use $y = 8 - x$ : $3 x - 2 (8 - x) = 4$ , so $5 x = 20$ .
$(4, 4)$
On the supplied grid, draw $y = x + 1$ and $y = 5 - x$ , then state their intersection.
3 marks
Both lines give $y = 3$ at $x = 2$ .
$(2, 3)$
Three adult tickets and two child tickets cost 46 dollars. One adult ticket and two child tickets cost 26 dollars. Find both prices.
3 marks
Subtract the equations to obtain two adult tickets costing 20 dollars.
Substitute the adult price into the second total.
Adult: 10 dollars; child: 8 dollars.

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 simultaneous equations and inequalities with graphical, algebraic and practical interpretations.

Official curriculum source ↗

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