Simultaneous Equations
Solving pairs of linear equations, plus one-linear-one-quadratic systems at Higher.
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Every point below, worked through in one video, and each point links straight to its own part.
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Simultaneous Equations, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Solve two linear simultaneous equations by elimination
Match the coefficients of one letter, then add or subtract the equations to eliminate it. For 3x + 2y = 12 and x − 2y = 4, the y terms already match: adding gives 4x = 16, so x = 4, then y = 0 by substituting back. Same signs subtract, opposite signs add.
Solve two linear simultaneous equations by substitution
Rearrange one equation to "y = …" (or "x = …") and substitute it into the other. From y = 2x − 1 and 3x + y = 14: 3x + 2x − 1 = 14, so x = 3 and y = 5. Substitution shines when one equation already has a letter on its own.
Solve simultaneous equations graphically, as the point where two lines cross
Each equation draws a line; the solution is the crossing point, because it is the only point on both. Plot both lines and read off the coordinates. Parallel lines never cross - that is the picture of simultaneous equations with no solution.
Set up simultaneous equations from worded problems
Two unknowns need two facts. "3 teas and 2 coffees cost £9.40; 1 tea and 2 coffees cost £6.20" becomes 3t + 2c = 9.4 and t + 2c = 6.2. Define your letters clearly, solve, and answer in the units the question asked for.
Solve one linear and one quadratic equation simultaneouslyHigher only
Substitute the linear equation into the quadratic one and solve the resulting quadratic. For y = x + 1 and x² + y² = 25: x² + (x + 1)² = 25 leads to 2x² + 2x − 24 = 0, giving x = 3 or x = −4. Each x needs its matching y - pair the answers as coordinates.
Common mistakes in Simultaneous Equations
The slips that cost marks year after year, and what to do instead.
Adding the equations when the matching terms have the same sign, so nothing cancels.
Same signs, subtract; different signs, add. Taking x + 2y = 8 away from 3x + 2y = 12 leaves 2x = 4, so x = 2 and y = 3.
Scaling only part of an equation, so 2x + y = 7 multiplied by 3 becomes 6x + 3y = 7.
Multiply every term on both sides, including the number on the right: 2x + y = 7 becomes 6x + 3y = 21.
Slipping on negatives when subtracting, so 5y − (−2y) comes out as 3y.
Taking away a negative is the same as adding: 5y − (−2y) = 7y. Write the subtraction out in brackets when the signs are mixed.
Stopping after the first unknown, or not checking the pair in the other equation.
Substitute to find the second letter, then test both values in the equation you did not use. With x = 2 and y = 3, 3x + 2y gives 12, as it should.
Mixing up the pairs, or finding only one y value, when solving a linear and a quadratic equation together.Higher only
Put each x back into the linear equation to find its partner. With y = x + 1 and x = 3 or x = −4, the answers are (3, 4) and (−4, −3).
Practice
Try a Simultaneous Equations question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Answer: y = 4
Doubling x + y = 6 gives 2x + 2y = 12; subtracting this from the first equation leaves y = 4.
- Multiply x + y = 6 by 2: 2x + 2y = 12.
- Subtract from 2x + 3y = 16: (2x + 3y) − (2x + 2y) = 16 − 12.
- This gives y = 4.
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