Algebra
Grade 5-8Simultaneous Equations
Simultaneous equations turn up every year on both Foundation and Higher tier GCSE Maths papers. This lesson covers everything you need: how to solve them by elimination and substitution, how to handle the harder linear-quadratic pairs that appear on Higher tier, and the mistakes that cost students the most marks.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
What are simultaneous equations?
Simultaneous equations are two (or more) equations that share the same unknowns, where you need to find values that satisfy both equations at the same time. Most GCSE questions give you two equations with two unknowns, usually x and y.
Graphically, solving a pair of linear simultaneous equations means finding the point where two straight lines cross. That is why there is usually exactly one solution, one point, one crossing.
Solving by elimination
Elimination means adding or subtracting the two equations so that one unknown cancels out. If the coefficients of x or y already match, you can add or subtract straight away. If they do not match, multiply one or both equations first so that they do, then add or subtract.
A common source of lost marks: when you multiply an equation to make coefficients match, every single term must be multiplied, including the number on its own on the right-hand side.
Solving by substitution
Substitution means rearranging one equation to make x or y the subject, then substituting that expression into the other equation. This is usually the faster method whenever one equation is already written as "y = ..." or "x = ...".
Linear and quadratic simultaneous equations (Higher tier)
Higher tier students also need to solve one linear equation together with one quadratic equation. The method is always substitution: rearrange the linear equation, substitute it into the quadratic equation, then solve the resulting quadratic.
Because you end up solving a quadratic, there are usually two pairs of solutions rather than one. Forgetting the second solution is one of the most common ways students drop marks on this topic.
Worked Examples
Three exam-style questions, fully solved.
Solve: 3x + y = 11 and x + y = 5
Easy- 1.The y terms already match, so subtract the second equation from the first: (3x + y) − (x + y) = 11 − 5
- 2.This gives 2x = 6, so x = 3
- 3.Substitute x = 3 into x + y = 5: 3 + y = 5, so y = 2
- 4.Check in the other equation: 3(3) + 2 = 11 ✓
Answer: x = 3, y = 2
Solve: 2x + 3y = 16 and x + y = 6
Medium- 1.The coefficients do not match, so multiply the second equation by 2: 2x + 2y = 12
- 2.Subtract this from the first equation: (2x + 3y) − (2x + 2y) = 16 − 12
- 3.This gives y = 4
- 4.Substitute y = 4 into x + y = 6: x = 2
- 5.Check: 2(2) + 3(4) = 4 + 12 = 16 ✓
Answer: x = 2, y = 4
Solve: y = x + 1 and x² + y² = 13
Hard- 1.Substitute y = x + 1 into the quadratic equation: x² + (x + 1)² = 13
- 2.Expand the bracket: x² + x² + 2x + 1 = 13
- 3.Simplify: 2x² + 2x − 12 = 0, then divide by 2: x² + x − 6 = 0
- 4.Factorise: (x + 3)(x − 2) = 0, so x = −3 or x = 2
- 5.Find y for each: if x = 2, y = 3. If x = −3, y = −2
- 6.Check both: 2² + 3² = 13 ✓ and (−3)² + (−2)² = 13 ✓
Answer: x = 2, y = 3 or x = −3, y = −2
Avoid These
The most common mistakes students make.
Only multiplying part of an equation when scaling it for elimination, forgetting to multiply the number on the right-hand side as well.
Sign errors when subtracting equations, especially forgetting that subtracting a negative term makes it positive.
Stopping after finding one pair of solutions on a linear-quadratic question, when a quadratic almost always gives two valid answers.
Substituting back into the equation you already used to find the first unknown, instead of double-checking in the other original equation.
Not checking the final answer in both original equations, which is the fastest way to catch an arithmetic slip before losing marks.
FAQ
Questions parents and students ask.
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