Algebra · GCSE Maths
Simultaneous equations
Solve two linear equations together by elimination or substitution, and read the solution as the intersection of two lines.
Two equations, one pair (x, y). Make a coefficient match, then add or subtract to eliminate a letter.
The important bits
What you need to know
- 1
A solution (x, y) must satisfy both equations at once. That pair is where the two lines cross.
- 2
Elimination: multiply one or both equations so that one letter has the same (or opposite) coefficient, then subtract or add.
- 3
If the matching coefficients have the same sign, subtract the equations. If they have opposite signs, add.
- 4
Substitution: rearrange one equation for a letter, then replace that letter in the other equation.
- 5
After finding x, substitute back into the simpler original equation to find y. Then check both originals.
- 6
Worded problems become simultaneous equations once you define two unknowns and write two true sentences about them.
- 7
A linear and a quadratic together: substitute the linear into the quadratic, then solve the resulting quadratic for x.
- 8
If elimination produces 0 = 0, the lines are the same. If it produces 0 = 4, the lines are parallel and there is no solution.
Go deeper
Elimination is organised cancellation
Take 2x + 3y = 16 and 5x − 3y = 5. The y coefficients are already opposites, so add: 7x = 21, x = 3. Substitute into the first equation: 6 + 3y = 16, 3y = 10, y = 10/3. When coefficients do not match, multiply. For x + 2y = 8 and 3x + 5y = 21, multiply the first by 3: 3x + 6y = 24. Subtract the second: y = 3, then x + 6 = 8, x = 2. Write “multiply (1) by 3” in the margin so the examiner can see the scale. Subtracting when you should add is the usual disaster: you double a term instead of cancelling it. Look only at the letter you are eliminating, and ask whether those two coefficients would cancel by adding or by subtracting.
Go deeper
Substitution when one equation is already simple
If y = 2x − 1 sits next to 3x + 4y = 11, do not eliminate. Replace y: 3x + 4(2x − 1) = 11, 3x + 8x − 4 = 11, 11x = 15, x = 15/11, then y follows. Substitution is also the method for a line and a curve. y = x + 2 and y = x² − 4 become x + 2 = x² − 4, so x² − x − 6 = 0, (x − 3)(x + 2) = 0. Two intersections are possible. Each x still needs its y from the linear equation, not from a mix of both. Graphically you are finding crossings; algebraically you are insisting that the same pair sits on both graphs. If the quadratic discriminant is negative, the line misses the curve.
See the idea in action
Solve 2x + y = 7 and 3x − 2y = 7. Multiply the first equation by 2: 4x + 2y = 14. Add to the second: 7x = 21, x = 3. From 2(3) + y = 7, y = 1. Check in the second equation: 9 − 2 = 7.
Exam technique
Turn knowledge into marks
Number the equations (1) and (2). After a multiplication, number the new line (3). Examiners follow labels more easily than a block of unmarked algebra.
Common mistakes
Do not give these marks away
- 01
Adding equations when the target coefficients already have the same sign, so the letter doubles instead of vanishing.
- 02
Finding x and then substituting into a rearranged line that still contains the eliminated letter wrongly copied.
- 03
Giving only x as the answer when the question asked for both values.
The solution of x + y = 5 and x − y = 1 is
Ax = 3, y = 2
Bx = 2, y = 3
Cx = 5, y = 1
Dx = 1, y = 5
Show the answer
x = 3, y = 2. Add the equations: 2x = 6, so x = 3. Then 3 + y = 5, y = 2. Subtracting instead would also work: 2y = 4, y = 2.
Quick questions
If this is the bit you searched
Should I always eliminate rather than substitute?
Eliminate when both equations are in the form ax + by = c. Substitute when one letter is already isolated, or when one equation is quadratic.
What if both letters vanish and I get 0 = 0?
The two equations describe the same line. There are infinitely many solutions, not a unique pair (x, y).