Simultaneous equations, finally easy
Learn to solve two unknowns with receipts. Use elimination, multiplication, and substitution. Try exam-style questions with answers.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (648 KB)How can you work out two prices when nobody tells you either one?
Simultaneous equations feel hard because there are two unknowns. Use the café receipts analogy to find the prices. Line up the receipts and subtract to find the difference.
The picture to keep in your head: Two café receipts
Two receipts from the same café: 2 coffees and a cake cost £7, 1 coffee and a cake cost £5. Line them up and the only difference is one extra coffee, which must cost the £2 difference. Subtracting the receipts makes the cake vanish; that is elimination. Multiplying a whole equation is buying the whole order twice: every item and the total double.
The method, every time
- Line up the receipts to see what’s different.
- Subtract receipts with same signs to remove the same items.
- Multiply a receipt to match the number of items in the other.
- Add receipts with different signs to cancel out the same items.
- Find one price first, then use it to find the other.
Words you'll see
- elimination
- Subtracting equations to remove one variable.
- multiplication
- Scaling an equation to match the other.
- substitution
- Using one equation to find the value of a variable.
- unknowns
- Variables you need to find, like coffee and cake prices.
- receipts
- Equations that show how much each item costs together.
Same sign? Subtract. Different signs? Add.
Imagine two café receipts. If the same item appears with the same sign in both, subtract them to cancel it out. This is like removing a charge with a refund. If the signs differ, add them to cancel. This rule helps you find the value of one item at a time.
Watch out
- Subtracting a negative wrongly. Remember subtracting a negative is adding the positive
- Forgetting to subtract the totals. Always subtract the entire equation, not just one term
- Stopping after finding x. Always find both variables to complete the solution
Multiply a whole receipt first
Imagine two café receipts. If you want to match the same item, you might need to buy the whole order again. That's multiplying a whole equation. Multiply one or both equations so one letter matches. Multiply every term and the total.
Watch out
- Multiplying the letters but not the total. Multiply every term in the equation, including the total
- Choosing the harder letter to match. Choose the letter that makes the math easier
When one equation says y = ...
Imagine two receipts from the same café. One says y = 3x - 4, the other says 2x + y = 11. Since y is already known in the first receipt, you can swap it into the second. Like swapping the cake for its price. This is substitution. Just plug the expression for y into the other equation and solve.
Watch out
- Dropping the brackets so the minus is lost. Keep brackets when substituting y
- Putting x back into the wrong equation. Always substitute x into the original y = 3x - 4
One straight line, one curve
Imagine two receipts: one has a straight line of items, the other a curved line of items. When a line crosses a curve, they share the same x and y values. To find where they meet, substitute the straight line into the curve. This gives you a quadratic equation. Solve it to find the x-values, then use the straight line to find the matching y-values. The rule is: substitute the straight line into the curve, rearrange to = 0, factorise, and find the x-values, then find the y-values.
Watch out
- Giving x values without their y values. Always give x and y values as pairs.
- Expanding (x + 1)2 as x2 + 1. Expand (x + 1)2 as x2 + 2x + 1.
23 exam-style simultaneous equations questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Same number in front: add or subtract
- 1Solve x + y = 10 and x - y = 4.[3 marks]
Show the answer
x = 7, y = 3
Add: 2x = 14. - 2Solve 3a + b = 11 and a + b = 5.[3 marks]
Show the answer
a = 3, b = 2
Subtract: 2a = 6. - 3Solve 2x + 3y = 19 and 2x + y = 9.[3 marks]
Show the answer
x = 2, y = 5
Subtract: 2y = 10. - 4Solve 4p + q = 15 and 4p - 2q = 6.[3 marks]
Show the answer
p = 3, q = 3
Subtract: 3q = 9. - 5Solve 5m + 2n = 24 and 3m - 2n = 8.[3 marks]
Show the answer
m = 4, n = 2
Add: 8m = 32.
B · Multiply first
- 6Solve 2x + y = 11 and x + 3y = 13.[4 marks]
Show the answer
x = 4, y = 3
(1) × 3 - (2): 5x = 20. - 7Solve 3x + 2y = 16 and x - y = 2.[4 marks]
Show the answer
x = 4, y = 2
(2) × 2 + (1): 5x = 20. - 8Solve 5a - 2b = 14 and 2a + b = 11.[4 marks]
Show the answer
a = 4, b = 3
(2) × 2 + (1): 9a = 36. - 9Solve 3x + 4y = 25 and 2x + 3y = 18.[4 marks]
Show the answer
x = 3, y = 4
(1) × 2 - (2) × 3: -y = -4. - 10Solve 5x - 3y = 1 and 2x + 5y = 19.[4 marks]
Show the answer
x = 2, y = 3
(1) × 5 + (2) × 3: 31x = 62. - 11Solve 4x + 3y = 5 and 3x - 2y = 8.[4 marks]
Show the answer
x = 2, y = -1
(1) × 2 + (2) × 3: 17x = 34.
C · Substitution
- 12Solve y = 2x + 1 and 3x + y = 16.[3 marks]
Show the answer
x = 3, y = 7
3x + 2x + 1 = 16, so 5x = 15. - 13Solve x = 3y - 2 and 2x + y = 17.[3 marks]
Show the answer
x = 7, y = 3
2(3y - 2) + y = 17, so 7y = 21. - 14Solve 0.5x + y = 7 and x - y = 2.[3 marks]
Show the answer
x = 6, y = 4
Add: 1.5x = 9.
D · Word problems
- 15Two adult and three child cinema tickets cost £36. One adult and two child tickets cost £21. Find the cost of each ticket.[4 marks]
Show the answer
Adult £9, child £6
2a + 3c = 36, a + 2c = 21; (2) × 2 - (1): c = 6. - 16A taxi charges a fixed fee plus a price per mile. A 5-mile journey costs £9.50 and an 8-mile journey costs £14. Find the fixed fee and the price per mile.[4 marks]
Show the answer
Fee £2, £1.50 a mile
f + 5p = 9.5, f + 8p = 14; subtract: 3p = 4.5. - 17A jar holds 25 coins, all 10p or 20p. Together they are worth £3.70. How many of each coin are there?[4 marks]
Show the answer
13 × 10p, 12 × 20p
t + w = 25, 10t + 20w = 370; (1) × 10 and subtract: 10w = 120. - 18Two numbers add up to 47. Their difference is 13. Find the two numbers.[4 marks]
Show the answer
30 and 17
x + y = 47, x - y = 13; add: 2x = 60. - 19Amara is three times as old as Ben. In 5 years' time she will be twice as old as him. How old are they now?[4 marks]
Show the answer
Ben 5, Amara 15
a = 3b and a + 5 = 2(b + 5), so 3b + 5 = 2b + 10.
E · Line and curve (Higher)
- 20Solve y = x + 1 and y = x² - 1.[5 marks]
Show the answer
x = 2, y = 3 and x = -1, y = 0
x² - 1 = x + 1, so x² - x - 2 = 0 = (x - 2)(x + 1). - 21Solve x² + y² = 25 and y = x + 1.[5 marks]
Show the answer
x = 3, y = 4 and x = -4, y = -3
x² + (x + 1)² = 25, so x² + x - 12 = 0 = (x + 4)(x - 3). - 22Solve y = 2x - 3 and xy = 2.[5 marks]
Show the answer
x = 2, y = 1 and x = -0.5, y = -4
x(2x - 3) = 2, so 2x² - 3x - 2 = 0 = (2x + 1)(x - 2). - 23The lines y = 2x + 1 and y = 7 - x cross at the point P. Find the coordinates of P.[5 marks]
Show the answer
P = (2, 5)
2x + 1 = 7 - x, so 3x = 6.
Simultaneous equations exam checklist
- Label equations (1) and (2) to keep track.
- Multiply every term in an equation to match the other.
- Same sign? Subtract. Different signs? Add. Make sure you get the right one.
- Find both unknowns and check in the other equation.
- Quadratics give two pairs of answers, so check both.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
Download the PDF (648 KB)Want maths to click like this?
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