Year 10 - Simultaneous equations - Free practice

Year 10 Simultaneous Equations Worksheets

Substitution, elimination, graphical - all three methods.

Quick answer

How do you solve simultaneous equations?

Simultaneous equations are two equations with two unknowns solved together to find values that satisfy both at once. Use substitution (rearrange one equation and sub into the other), elimination (add or subtract to remove one variable), or find the intersection point on a graph. For example, x + y = 5 and x - y = 1 gives x = 3, y = 2.

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Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

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Worked examples

Year 10 simultaneous equations: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Solving by substitution
Solve the simultaneous equations:
$\begin{gathered}y = 2x - 1 \\ 3x + y = 9\end{gathered}$
Show worked solution
Key idea: Substitute the expression for $y$ from the first equation directly into the second equation.
1. Substitute $y = 2x - 1$ into $3x + y = 9$ :
  $3x + (2x - 1) = 9$
2. Simplify and solve for $x$ :
  $\begin{aligned}5x - 1 &= 9 \\ 5x &= 10 \\ x &= 2\end{aligned}$
3. Substitute $x = 2$ back into $y = 2x - 1$ :
  $y = 2(2) - 1 = 3$
4. Check in $3x + y = 9$ : $3(2) + 3 = 9$ ✓
Answer: $x = 2,\; y = 3$
Example 2 · Solving by elimination
Solve the simultaneous equations:
$\begin{gathered}3x + 2y = 16 \\ 5x - 2y = 8\end{gathered}$
Show worked solution
Key idea: Add the equations to eliminate $y$ , since the $y$ -coefficients are opposites.
1. Add the two equations:
  $\begin{aligned}(3x + 2y) + (5x - 2y) &= 16 + 8 \\ 8x &= 24 \\ x &= 3\end{aligned}$
2. Substitute $x = 3$ into $3x + 2y = 16$ :
  $\begin{aligned}9 + 2y &= 16 \\ 2y &= 7 \\ y &= 3.5\end{aligned}$
3. Check in $5x - 2y = 8$ : $5(3) - 2(3.5) = 15 - 7 = 8$ ✓
Answer: $x = 3,\; y = 3.5$
Example 3 · Setting up and solving a word problem
Two friends, Aiden and Brianna, buy items at a school canteen. Aiden buys 2 pies and 1 drink and pays $8.70. Brianna buys 1 pie and 1 drink and pays $5.20. Find the cost of one pie and one drink.
Show worked solution
Key idea: Define variables, write two equations from the given information, then solve.
1. Let $p$ = cost of one pie (dollars) and $d$ = cost of one drink (dollars). Write the equations:
  $\begin{aligned}2p + d &= 8.70 (1)\\ p + d &= 5.20 (2)\end{aligned}$
2. Subtract equation (2) from equation (1) to eliminate $d$ :
  $\begin{aligned}(2p + d) - (p + d) &= 8.70 - 5.20 \\ p &= 3.50\end{aligned}$
3. Substitute $p = 3.50$ into equation (2):
  $\begin{aligned}3.50 + d &= 5.20 \\ d &= 1.70\end{aligned}$
4. Check in equation (1): $2(3.50) + 1.70 = 7.00 + 1.70 = 8.70$ ✓
Answer: One pie costs $3.50 and one drink costs $1.70.

Common mistakes

Where students usually lose marks

Substituting into the same equation

Found $x = 2$ from equation (1), then substituted back into equation (1) to find $y$ , instead of using equation (2).

You can substitute back into either equation to find the second variable, but always check your answer in the other equation. Using a different equation from the one you solved reduces the chance of carrying an error forward.

Sign errors when subtracting equations

$(3x + 2y) - (x - 2y) = 2x$ instead of $2x + 4y$ .

When subtracting an equation, every term on both sides changes sign. Write out the subtraction line by line, or multiply the equation by $-1$ first and then add, to keep track of signs.

Forgetting to multiply every term when scaling an equation

Multiplying $3x - y = 5$ by $2$ to get $6x - y = 10$ instead of $6x - 2y = 10$ .

When you multiply an equation by a number, every single term on both sides must be multiplied. Check each term one at a time before moving on.

Stopping after finding only one variable

Solved $x = 4$ and wrote the answer as $x = 4$ , without finding $y$ .

Simultaneous equations always require both variables. Once you find one, substitute it back in to find the other, and state both values clearly in your answer.

Try one yourself

Your turn

Solve the following pair of simultaneous equations using any method.

$3x + 2y = 16$
$x - y = 2$

Find the value of $x$ .

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About this worksheet

Why we built it

Simultaneous equations are one of the most-tested Year 10 topics. These worksheets give your student practice on all three solution methods - substitution, elimination, graphical - plus applied problems that need the student to set up the system before solving.

What's covered

Sub-skills your student will practise

✓Solving by substitution
✓Solving by elimination
✓Solving graphically (intersection of two lines)
✓Recognising no solution / infinite solutions
✓Word problems requiring setup of two equations

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Frequently asked

Questions parents ask about simultaneous equations

What are simultaneous equations?

Two or more equations that students need to solve together to find values that satisfy all of them. Most Year 10 work is two equations with two variables (x and y).

What are the three solution methods?

Substitution (solve one equation for one variable, then sub into the other), elimination (add or subtract the equations to cancel a variable), and graphical (sketch both lines and find the intersection).

Which method should students use?

Substitution if one equation is already 'y = ...' or 'x = ...'. Elimination if the coefficients line up neatly. Graphical when the question specifically asks for it or as a quick check.

When do simultaneous equations have no solution?

When the two lines are parallel (same gradient, different y-intercept). When they're the same line (same gradient and y-intercept), there are infinitely many solutions. Year 10 expects students to recognise both cases.

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