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What are simultaneous equations, and how do you solve them?

Quick answer

Simultaneous equations (a system of equations in the US) are two or more equations that share unknowns and must all be true at once. Solve them by elimination, substitution or graphing.

Ages 13–16Secondary

In one sentence

Simultaneous equations are two (or more) equations with the same unknowns, and solving them means finding the values that make every equation true at the same time.

Explained

One equation with two unknowns, like x + y = 10, has endless answers: 1 and 9, 4 and 6, 2.5 and 7.5. Add a second clue, x − y = 4, and only one pair works for both: x = 7 and y = 3. That is the whole idea. Each equation is a clue, and you need as many clues as unknowns.

In the UK and Australia these are called simultaneous equations ("simultaneous" means "at the same time"). In the US the same thing is called a system of equations or a system of linear equations. The methods are identical.

  • apple
  • banana
  • pencil
2 apples and 1 banana cost £1.30; 1 apple and 1 banana cost 80p. Two clues, two unknowns.

There are three main methods.

Elimination. Add or subtract the equations so that one letter disappears. Take 2a + b = 130 and a + b = 80 (prices in pence). Subtract the second from the first: the b's cancel and you get a = 50. Put that back into a + b = 80 to get b = 30. An apple costs 50p and a banana 30p. If the numbers in front of a letter don't match, multiply a whole equation first so that they do.

Substitution. If one equation already says what a letter equals, swap it into the other. With y = 3x and x + y = 8, replace y by 3x: x + 3x = 8, so 4x = 8, x = 2 and y = 6.

Graphing. Every linear equation is a straight line (see line graphs). The solution is the point where the two lines cross. y = x + 1 and y = −x + 5 cross at (2, 3), so x = 2 and y = 3. Graphing is great for seeing what is going on, but it can be hard to read exact answers, so algebra is used to finish the job.

Two lines can also be parallel (same slope, never meet: no solution) or be the same line (every point works: infinitely many solutions). Most school questions have exactly one solution. Simultaneous equations build on solving single equations and on plotting points with coordinates.

Examples

Equations Best method Working Solution
x + y = 10, x − y = 4 add 2x = 14 x = 7, y = 3
y = 3x, x + y = 8 substitute 4x = 8 x = 2, y = 6
2x + y = 11, x + y = 7 subtract x = 4 x = 4, y = 3
3x + 2y = 12, x + 2y = 8 subtract 2x = 4 x = 2, y = 3
2x + 3y = 13, 5x − y = 7 multiply the 2nd by 3, then add 17x = 34 x = 2, y = 3

Worked example (last row): multiply 5x − y = 7 by 3 to get 15x − 3y = 21. Now −3y and +3y have different signs, so add: (2x + 15x) + (3y − 3y) = 13 + 21, which is 17x = 34, so x = 2. Then 2(2) + 3y = 13, so 3y = 9 and y = 3. Check in the other equation: 5(2) − 3 = 7. Correct.

Common mistakes

  • Adding when you should subtract. Fix: remember "same signs subtract, different signs add". +2y and +2y: subtract. +2y and −2y: add.
  • Multiplying only part of an equation. If you multiply by 3, every term changes, including the number on the right: 5x − y = 7 becomes 15x − 3y = 21, not 15x − 3y = 7.
  • Sign slips when subtracting. 4y − (−2y) = 6y, not 2y. Write the brackets in.
  • Stopping after the first letter. You need both values. Then check them in both original equations.

Step by step

Step by step

  1. Line them upWrite both equations as ax + by = c, with x under x, y under y and the numbers on the right.
  2. Match a coefficientChoose a letter to remove. If its numbers differ, multiply one or both equations so they match (ignoring the sign).
  3. EliminateSame signs: subtract the equations. Different signs: add them. One letter disappears.
  4. SolveSolve the simple equation that is left for the remaining letter.
  5. Substitute backPut that value into either original equation to find the other letter.
  6. CheckPut both values into the other equation. If it works, you are done.

For substitution, steps 2–3 change: rearrange one equation to "y = …" (or "x = …"), then replace that letter in the other equation.

Practice questions

Solve the simultaneous equations

Try each one, then check your answer.

  1. x + y = 9 and x − y = 3
    Show answer
    Add: 2x = 12, so x = 6; then y = 9 − 6 = 3. x = 6, y = 3
  2. x + y = 15 and x − y = 5
    Show answer
    Add: 2x = 20, so x = 10; y = 5. x = 10, y = 5
  3. y = 2x and x + y = 15
    Show answer
    x + 2x = 15, so 3x = 15. x = 5, y = 10
  4. y = x + 3 and 2x + y = 18
    Show answer
    2x + x + 3 = 18, so 3x = 15. x = 5, y = 8
  5. 2x + y = 13 and x + y = 8
    Show answer
    Subtract: x = 5; then y = 8 − 5. x = 5, y = 3
  6. 3x + y = 14 and x − y = 2
    Show answer
    Add: 4x = 16, so x = 4; 4 − y = 2. x = 4, y = 2
  7. 4x + 3y = 25 and 2x + 3y = 17
    Show answer
    Subtract: 2x = 8, so x = 4; 3y = 17 − 8 = 9. x = 4, y = 3
  8. 5x + 2y = 19 and 3x − 2y = 5
    Show answer
    Add: 8x = 24, so x = 3; 15 + 2y = 19, so 2y = 4. x = 3, y = 2
  9. x + 2y = 11 and 3x + y = 13
    Show answer
    ×3 the first: 3x + 6y = 33. Subtract: 5y = 20, so y = 4; x = 11 − 8. x = 3, y = 4
  10. 2x + 3y = 12 and 3x + 2y = 13
    Show answer
    ×3 and ×2: 6x + 9y = 36 and 6x + 4y = 26. Subtract: 5y = 10, so y = 2; 2x = 6. x = 3, y = 2
  11. 3x − 2y = 4 and 2x + 5y = 9
    Show answer
    ×2 and ×3: 6x − 4y = 8 and 6x + 15y = 27. Subtract: 19y = 19, so y = 1; 3x = 6. x = 2, y = 1
  12. y = 3x − 1 and y = x + 5
    Show answer
    3x − 1 = x + 5, so 2x = 6. x = 3, y = 8
  13. x + y = 2 and 2x − y = 10
    Show answer
    Add: 3x = 12, so x = 4; y = 2 − 4. x = 4, y = −2
  14. How many solutions do y = 3x + 2 and 6x − 2y = 4 have?
    Show answer
    The second is y = 3x − 2: same slope, different intercept, so the lines are parallel. No solution
  15. How many solutions do x + y = 4 and 2x + 2y = 8 have?
    Show answer
    The second is just the first doubled, so it is the same line. Infinitely many

Word problems

Simultaneous equations in real life

Try each one, then check your answer.

  1. Two numbers add up to 30 and their difference is 8. What are they?
    Show answer
    x + y = 30, x − y = 8. Add: 2x = 38. 19 and 11
  2. 3 pens and 2 notebooks cost £8. 1 pen and 2 notebooks cost £5. Find each price.
    Show answer
    Subtract: 2 pens = £3, so a pen is £1.50; 2 notebooks = £5 − £1.50 = £3.50. Pen £1.50, notebook £1.75
  3. 2 adult and 3 child cinema tickets cost £38. 1 adult and 2 child tickets cost £22. Find each price.
    Show answer
    Double the second: 2a + 4c = 44. Subtract the first: c = 6; a = 22 − 12. Adult £10, child £6
  4. A farm has chickens and cows: 20 heads and 56 legs. How many of each?
    Show answer
    c + w = 20 and 2c + 4w = 56. Halve the second: c + 2w = 28. Subtract: w = 8. 12 chickens, 8 cows
  5. Gym A charges a £20 joining fee plus £3 a visit. Gym B charges £5 a visit. After how many visits do they cost the same?
    Show answer
    20 + 3x = 5x, so 2x = 20. 10 visits (both cost £50)

Questions people ask

What is a system of equations?

It is the US name for simultaneous equations: a set of equations with the same unknowns that must all be true together. The solution is the set of values that satisfies every equation.

How do you solve simultaneous equations by elimination?

Make the number in front of one letter the same in both equations, then add (different signs) or subtract (same signs) to remove it. Solve for the other letter, substitute back, and check.

How do you solve simultaneous equations by substitution?

Rearrange one equation into "y = …" or "x = …", then put that expression into the other equation. You get one equation with one unknown, which you solve as normal.

How do you solve simultaneous equations graphically?

Draw both lines on the same axes and read off the point where they cross. On a graphing tool such as Desmos, type both equations and tap the intersection point. Exact answers still come from algebra.

Can simultaneous equations have no solution?

Yes. If the lines are parallel (same slope, different intercept) there is no solution. If both equations describe the same line there are infinitely many.

How do you solve simultaneous equations with 3 variables?

You need three equations. Use elimination to remove one letter twice, which leaves two equations in two letters; solve those, then substitute back. Later on, matrices do the same job in a tidy way.

Can I use a calculator?

Some scientific calculators have an equation mode that solves systems, and online solvers exist. They are useful for checking, but exams usually ask you to show the working, so practise the method first.

Quick quiz

Simultaneous equations (systems of equations)

Check what you know with 5 quick questions.

Question 1 of 5

Which pair solves x + y = 10 and x − y = 4?

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