Simultaneous equations: how to solve them step by step

What are simultaneous equations?

Simultaneous equations are a pair of equations that share the same unknowns. "Simultaneous" means "at the same time": the solution makes both true together.

One equation with two unknowns, like x + y = 10, has lots of answers: 1 and 9, 2 and 8, and so on. A second equation, x − y = 4, narrows it down to x = 7 and y = 3. Each method below turns the pair into one equation with one unknown, which you solve as in solving equations. Then put both values back into both equations to check.

How to solve simultaneous equations by elimination

To solve simultaneous equations by elimination, add or subtract the two equations so that one letter disappears. This works when that letter has the same number in front of it (the same coefficient) in both.

• If the signs are different (+y in one, −y in the other), add the equations.
• If the signs are the same (+2x in both), subtract one equation from the other.

Worked example: adding to eliminate

Solve 3x + y = 11 and 2x − y = 4.

1. The y terms have different signs, so add the equations. The y terms cancel, and 11 + 4 = 15, so 5x = 15.
2. Divide both sides by 5: x = 3.
3. Put x = 3 into the first equation: 9 + y = 11, so y = 2.
4. Check in the second equation: 2 × 3 = 6, and 6 − 2 = 4. It works, so x = 3 and y = 2.

Worked example: subtracting to eliminate

Solve 2x + 3y = 12 and 2x + y = 8.

1. Both equations have +2x, so subtract the second from the first. 3y − y = 2y, and 12 − 8 = 4. So 2y = 4, and y = 2.
2. Put y = 2 into the second equation: 2x + 2 = 8, so 2x = 6 and x = 3.
3. Check in the first equation: 2 × 3 = 6, 3 × 2 = 6, and 6 + 6 = 12. So x = 3 and y = 2.

Worked example: multiplying first

If neither letter matches, multiply one or both equations first. Solve 2x + 3y = 13 and 5x − 2y = 4.

1. Make the y terms match. Multiply every term of the first equation by 2: 4x + 6y = 26. Multiply every term of the second by 3: 15x − 6y = 12.
2. The signs of 6y are different, so add: 19x = 38, which gives x = 2.
3. Put x = 2 into the first equation: 4 + 3y = 13, so 3y = 9 and y = 3.
4. Check in the second equation: 5 × 2 = 10, 2 × 3 = 6, and 10 − 6 = 4. So x = 2 and y = 3.

How to solve simultaneous equations by substitution

To solve simultaneous equations by substitution, rearrange one equation to say "y = ..." (or "x = ..."), then put that expression into the other equation in place of the letter. It's quickest when one equation already starts y = or x =.

Worked example: substitution

Solve y = 2x + 1 and 3x + y = 16.

1. Write 2x + 1 in place of y in the second equation: 3x + 2x + 1 = 16.
2. Collect like terms: 5x + 1 = 16, so 5x = 15 and x = 3.
3. Put x = 3 into y = 2x + 1: 2 × 3 = 6, and 6 + 1 = 7, so y = 7.
4. Check in the second equation: 3 × 3 = 9, and 9 + 7 = 16. So x = 3 and y = 7.

Simultaneous equations from a word problem

GCSE asks students to write two simultaneous equations from a situation, solve them and interpret the answer.

Example: 2 adult tickets and 3 child tickets cost £26. 1 adult ticket and 2 child tickets cost £15. How much is each ticket?

Call the adult price a and the child price c. Then 2a + 3c = 26 and a + 2c = 15. Multiply the second equation by 2: 2a + 4c = 30. Both now have +2a, so subtract the first equation: c = 4. Then a + 8 = 15, so a = 7. An adult ticket is £7 and a child ticket is £4.

Check: 2 × 7 = 14, 3 × 4 = 12, and 14 + 12 = 26.

How to solve quadratic simultaneous equations

To solve quadratic simultaneous equations, where one equation is linear and the other has an x² in it, substitute the linear equation into the quadratic one, then solve the quadratic. There are usually two pairs of answers. This is higher tier GCSE content: the DfE's subject content puts it in bold, for the more highly attaining students.

Worked example: a line and a curve

Solve y = x² and y = x + 6.

1. Both equations say what y is, so x² = x + 6.
2. Rearrange so one side is 0: x² − x − 6 = 0.
3. Factorise the quadratic: (x − 3)(x + 2) = 0, so x = 3 or x = −2.
4. Find y for each with y = x + 6, keeping each x with its own y: x = 3, y = 9 and x = −2, y = 4.
5. Check both pairs in y = x²: 3² = 9, and −2 squared is 4. Both work.

A line and a circle

Higher tier questions also use a circle with its centre at the origin, such as x² + y² = 25. With y = x + 1, substituting gives x² + (x + 1)² = 25. That simplifies to x² + x − 12 = 0, so x = 3, y = 4 or x = −4, y = −3. Check: 3² + 4² = 25. When a quadratic doesn't factorise, students use the quadratic formula: see solving quadratic equations.

How to solve simultaneous equations graphically

To solve simultaneous equations graphically, draw the graph of each equation on the same axes. The point where they cross is the solution. The lines for x + y = 10 and x − y = 4 cross at (7, 3), so x = 7 and y = 3. Both key stage 3 and GCSE ask for approximate solutions read from graphs like this.

A line can cross a curve twice, touch it once or miss it. The line y = x + 6 crosses the curve y = x² at (−2, 4) and (3, 9), the two answers above. Graphs only give approximate answers, which is why the algebra matters.

Common mistakes with simultaneous equations

• Adding when you should subtract. Look at the letter you want to get rid of: same signs, subtract; different signs, add.
• Multiplying only part of an equation. Every term, on both sides, gets multiplied, just as in expanding brackets.
• Sign slips when subtracting. Taking away −2y is the same as adding 2y. The negative numbers guide covers the rules.
• Stopping after one letter, or mixing up the pairs in a quadratic question. Each x has its own y.
• Not checking. Putting both values into both equations catches most slips.

How to help at home

• Start with puzzles. "Two numbers add to 15 and one is 3 more than the other." Guess first, then see how the algebra gets there.
• Ask which method and why. If one equation starts y =, substitute. If a letter already matches, eliminate.
• Keep times tables quick. Finding a number that both coefficients go into, like 6 for 2y and 3y, uses multiples all the time.
• Ask the school what's coming. Schools choose the order of GCSE topics across Years 10 and 11, so their scheme of work shows when simultaneous equations come up.

Where simultaneous equations fit

Key stage 3, from Year 7 to Year 9, builds the groundwork: solving equations with one unknown, algebra basics such as substituting, and approximate solutions of simultaneous linear equations from graphs. Key stage 4, in Years 10 and 11, adds solving two linear simultaneous equations with algebra, on both GCSE tiers. Linear and quadratic pairs are higher tier only. See GCSE maths foundation, GCSE maths higher and KS3 maths for the other topics.

Simultaneous equations questions to try

1. Solve x + y = 12 and x − y = 2.
2. Solve 2x + y = 11 and x + y = 7.
3. Solve 4x + 3y = 26 and 2x + y = 10.
4. Solve y = 3x − 2 and 2x + y = 13 by substitution.
5. Two numbers add up to 20, and one is 6 more than the other. Write two equations and find the numbers.
6. (Higher tier) Solve y = x² and y = 2x + 3.

Answers

1. x = 7, y = 5 (add the equations: 2x = 14)
2. x = 4, y = 3 (subtract the equations: x = 4)
3. x = 2, y = 6 (multiply the second by 3 to get 6x + 3y = 30, then subtract the first: 2x = 4)
4. x = 3, y = 7 (2x + 3x − 2 = 13, so 5x = 15)
5. x + y = 20 and x − y = 6, so the numbers are 13 and 7.
6. x = 3, y = 9 and x = −1, y = 1 (x² − 2x − 3 = 0 factorises to (x − 3)(x + 1) = 0)