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Simultaneous Equations: Free Printable Worksheet for Year 9
- ForYear 9
- StageKey Stage 3
- CostFree
- FormatPrintable PDF
Simultaneous Equations is a free printable maths worksheet for Year 9 (ages 13 to 14). Children use elimination to find the values of x and y. They add or subtract a pair of equations to remove one letter, solve for the other, then put it back to find the first. Page 2 gives the grown-up notes, help, a stretch idea and the answers.
It comes from the Mini Minds Hub, the free library of printables from Jaques of London, the British games and toy company that has been making games for families since 1795. Print both pages, give page 1 to the student and keep page 2 for yourself. Black and white printing works well.
What is on the sheet
Each question gives a pair of equations in x and y. Children label them A and B, then decide whether to add or subtract to remove one letter. They solve for the letter that is left, put that value back into an equation to find the other, and check in both. The pairs start gently, then need more thought about which letter to remove. There is space to write under each question.
- x + y = 11 and x - y = 3
- x + y = 9 and x - y = 1
- 2x + y = 13 and x + y = 8
- 3x + y = 14 and x + y = 6
- x + 2y = 11 and x + y = 7
- 2x + 3y = 18 and 2x + y = 10
- 3x + 2y = 18 and x + 2y = 10
- 5x + y = 23 and 2x + y = 11
A worked example
A: 4x + y = 31 and B: x + y = 19. Find x and y.
This is a different pair from the ones on the sheet. It shows the same method with new numbers. Adding is used when one letter has opposite signs in the two equations, for example y in one and minus y in the other. Here the y terms match, so we subtract.
Method 1: subtract, then use B- Label the equations. A is 4x + y = 31. B is x + y = 19.
- The y terms are the same, so subtract B from A. 4x minus x is 3x. 31 minus 19 is 12.
- So 3x = 12, which gives x = 4.
- Put x = 4 into B: 4 + y = 19, so y = 15.
- Put x = 4 into A instead: 16 + y = 31, so y = 15.
- Both equations give the same y, which is a good sign.
- Check in B: 4 + 15 = 19. This is correct.
What does simultaneous mean?
Simultaneous equations are two or more equations that are true at the same time. The letters must stand for the same numbers in every equation. One equation with two unknown letters has many possible answers. A second equation narrows these down, and often leaves just one pair of values that works for both.
Elimination is one way to solve them. If a letter has the same size in both equations, adding or subtracting the equations removes it. That leaves a short equation with one letter, which students already know how to solve. Another method is substitution. Elimination is often quicker when the letters line up neatly.
Skills and curriculum link
- Solving linear equations
- Elimination
- Choosing to add or subtract
- Substituting values
- Checking answers
- Setting out working
This sheet supports the area below. It is matched to the published England framework. Please check it against your own school plan.
| Framework | Subject or area | Strand | Year |
|---|---|---|---|
| National Curriculum (England) | Maths | Algebra | Year 9 |
What children will be able to do
- I can add or subtract two equations to remove a letter.
- I can find the value of both letters.
- I can check my answers in the original equations.
Ideas that help
- Same letter, same valueIn a pair of simultaneous equations, x means the same number in both equations, and so does y.
- Equations stay balancedAdding or subtracting two true equations gives another true equation. Every term on both sides must be included.
- Look at the signsIf a letter has the same sign in both equations, subtracting removes it. If the signs are opposite, adding removes it.
- Checking is part of the methodPut both values into each original equation. If either one fails, look for a slip with signs or arithmetic.
Key words
- EquationA statement with an equals sign, saying two sides have the same value.
- SimultaneousTrue at the same time. Both equations must work for the same x and y.
- EliminateRemove. We make one letter disappear so we can solve for the other.
- SubstituteSwap a letter for its value. We put a number in place of x or y.
- CoefficientThe number in front of a letter. In 5x, the coefficient is 5.
- VariableA letter that stands for a number we are finding, such as x or y.
How to use it
- Print both pages. Page 1 is for the student. Keep page 2, which has the notes and answers.
- Write A and B beside the two equations in the first question. Look at the y terms and the x terms. Which letter can be removed?
- Add or subtract the equations, then solve the short equation that is left. Write each step on a new line.
- Put the value back into one equation to find the other letter. Then check both values in the other equation.
- Compare with page 2. Ask, "Why did you add or subtract here?" Come back another day to redo any that felt tricky.
Help, stretch and game ideas
If it feels too hard
Work the first pair together and write each step on its own line. Underline the matching letter in both equations. Ask whether the signs are the same or opposite. Always do the check at the end, so a slip is spotted early.
To stretch confident children
Choose your own values for x and y. Write two equations that both work with them, so that one letter can be eliminated. Swap with a partner and solve each other's pairs. Then try making a pair where subtracting is the better choice.
Make it a game
Play Secret Pair. One player secretly picks values for x and y and writes two equations that fit. The other player solves them to find the secret values. Swap roles and check each other's working. Score a point for each correct pair.
Mistakes to watch for
Watch forA student subtracts only some of the terms, or forgets to subtract the right hand sides.
Try thisWrite the subtraction in a column, with the equations lined up under each other. Do the x terms, the y terms and the numbers on the right, one at a time.
Watch forA student adds when they should subtract, so the letter does not disappear.
Try thisCheck the signs first. Same signs means subtract. Opposite signs means add. If the letter is still there after the step, try the other operation.
Watch forA student finds one letter and stops, thinking the question is finished.
Try thisRemind them that there are two unknowns. Ask, "What is the value of the other letter?" Then put the first value back into an equation.
Talk about it
- How do you decide whether to add or subtract the equations?
- What does it tell you if the check does not work?
- Could you solve this pair in a different way? What would you do first?
- Why must the same x and y work in both equations?
- How would you explain elimination to a younger student?
- What would you change in an equation so that a letter could be eliminated?
Take it off the page
Secret number clues
Pick two secret numbers and write two clues linking them, such as their sum and their difference. Ask someone at home to find the numbers. Then check them against both clues.
Equation of the day
Write one pair of equations each day on a scrap of paper. Solve it, check it and fold it into a pot. At the end of the week, pull out a few to redo for practice.
Swap and mark
Write three pairs of your own with known answers. Swap with a friend, solve each other's and mark with a coloured pencil. Talk about which pair was hardest to write.
Who it is for
Parents
You do not need to be a maths expert. Sit beside your child for the first pair and ask them to explain each step. Praise careful working and checking. If they get stuck, ask whether the signs match.
Teachers
Use it as a starter or plenary after teaching elimination, as a homework task or as an early finisher sheet. Page 2 has support and stretch ideas for mixed ability groups. It needs no preparation beyond printing.
Home educators
Print it as part of a wider algebra plan. It pairs well with equation practice from your main resource, and the sheet can be reused for revision or for a sibling later. Ask for written working, not only answers.
Where this fits
Before simultaneous equations, students solve one step and two step equations with a single letter. They also substitute numbers into expressions and formulae. These skills matter here, because every pair is reduced to an equation with one unknown.
After this, students meet pairs where one equation must be multiplied first so that a letter can be eliminated. They also learn the substitution method and see how each equation is a line on a graph. Harder forms follow in Key Stage 4.
Check it worked
Ask these when the sheet is finished. Listen for the reasoning, not just the answer.
- Why did you choose to add rather than subtract?
- How did you find the second letter?
- How can you check that both answers work?
Questions parents and teachers ask
- What age is Simultaneous Equations for?
It is written for Year 9 (ages 13 to 14), in Key Stage 3. It suits students who can already solve an equation with one letter and want practice with a pair of equations.
- What is elimination?
Elimination means adding or subtracting two equations so that one letter disappears. That leaves one equation with one letter, which can be solved. The value is then put back into an equation to find the other letter.
- How do I know whether to add or subtract?
Look at the letter you want to remove. If it has the same sign in both equations, subtract. If the signs are opposite, add. After the step, that letter should be gone.
- What if the answers do not check?
Check means putting both values into each original equation. If one fails, look for a sign slip or a missing term in the add or subtract step. Redo that line, then check again.
- Do I need page 2?
Page 2 is for the grown-up. It has notes on using the sheet, help for students who find it hard, a stretch idea and the answers. Keep it back until the student has tried and checked their own work.
- Can teachers use it in class?
Yes. It works as a starter, plenary, homework task, early finisher activity or table task. It needs no preparation beyond printing, and page 2 gives support and stretch ideas for mixed groups.
- Is it free and screen free?
Yes. It is free to print for home and classroom use. It needs only paper and a pencil, and it prints well in black and white on plain A4 paper.
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Made by Jaques of London, making games for families since 1795. A printable supports classroom learning and does not replace a teacher.