Simultaneous equations by graphs, substitution and elimination | Year 10

01 / LEARN

Understand the idea.

A pair must satisfy both equations

A simultaneous solution is an ordered pair that makes both equations true at once. Graphically it lies on both lines, at their intersection. A plotted intersection suggests the coordinates; substitution checks whether the values are exact rather than a reading estimate.

Choose a method that fits

Use substitution when one variable is already isolated: replace that variable in the other equation with its equal expression. This leaves one equation in one variable. After solving it, substitute back to find the second coordinate.

Use elimination when matching coefficients are easy to create. Multiply a whole equation by a nonzero number, then add or subtract the equations so one variable cancels. Find both variables and check the ordered pair in both original equations.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read and check an intersection

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

The lines y equals two x plus one and y equals seven minus x on equal-scale axes from zero to eight. The curves intersect at their common solution. Each line is identified by its equation; no extra solution marker is added.

Worked solution

1
Equate the two expressions for $y$ .
$2 x + 1 = 7 - x$
At an intersection, both lines have the same $x$ coordinate and the same $y$ value.
2
Solve for the common $x$ coordinate.
$\begin{aligned} 3 x + 1 & = 7 \\ 3 x & = 6 \\ x & = 2 \end{aligned}$
Add $x$ to both sides, subtract $1$ , then divide by $3$ .
3
Find the corresponding $y$ coordinate.
$y = 2 (2) + 1 = 5$
The first line supplies $y$ once $x$ is known.
4
Check the pair in the second equation.
$5 = 7 - 2$
The pair $(2, 5)$ satisfies both equations, so it is their intersection.

Answer

$(x, y) = (2, 5)$

EXAMPLE 02

Eliminate a variable

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

Worked solution

1
Create opposite coefficients for $y$ .
$\begin{aligned} 4 x + 6 y & = 26 \\ 9 x - 6 y & = 0 \end{aligned}$
Multiply the first original equation by $2$ and the second by $3$ .
2
Add the equations and solve for $x$ .
$\begin{aligned} 13 x & = 26 \\ x & = 2 \end{aligned}$
The $6 y$ and $- 6 y$ cancel when the equations are added.
3
Substitute into the second original equation.
$\begin{aligned} 3 (2) - 2 y & = 0 \\ - 2 y & = - 6 \\ y & = 3 \end{aligned}$
Using $x = 2$ leaves a linear equation for $y$ .
4
Verify both original equations.
$\begin{aligned} 2 (2) + 3 (3) & = 13 \\ 3 (2) - 2 (3) & = 0 \end{aligned}$
Both equalities hold, establishing $(x, y) = (2, 3)$ as the solution.

Answer

$(2, 3)$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$(3, 5)$

Substitute $y = x + 2$ into $x + y = 8$ : $2 x + 2 = 8$ , so $x = 3$ and $y = 5$ . The pair gives $5 = 3 + 2$ and $3 + 5 = 8$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

A pair must satisfy both equations

Choose a method that fits

See the method in action.

Read and check an intersection

Equate the two expressions for y.

Solve for the common x coordinate.

Find the corresponding y coordinate.

Check the pair in the second equation.

Eliminate a variable

Create opposite coefficients for y.

Add the equations and solve for x.

Substitute into the second original equation.

Verify both original equations.

Try it yourself.

Check your understanding

Keep practising

Equate the two expressions for $y$ .

Solve for the common $x$ coordinate.

Find the corresponding $y$ coordinate.

Create opposite coefficients for $y$ .

Add the equations and solve for $x$ .