M16 · Simultaneous equations | Edexcel GCSE Maths

Revise the key ideas

Simultaneous equations must both be true for the same pair of values. Their solution is the intersection of their graphs.
Intersection of x plus y equals seven and x minus y equals one
Enlarge diagram: Intersection of x plus y equals seven and x minus y equals one
Elimination adds or subtracts equations to remove a variable. If its coefficients are equal, subtract; if they are opposites, add.
Worked example: x + y = 7 and x − y = 1. Add to get 2x = 8, so x = 4. Substitute into x + y = 7 to obtain y = 3.
Worked example: 2x + 3y = 13 and 3x + 2y = 12. Multiply the first by 3 and the second by 2 to obtain 6x + 9y = 39 and 6x + 4y = 24. Subtract: 5y = 15, so y = 3 and x = 2.
When multiplying an equation, multiply every term on both sides. Keep negative signs and the equals sign aligned when eliminating.
Substitution is useful if one variable is already isolated. From y = 2x + 1 and x + y = 10, substitute: x + 2x + 1 = 10, giving x = 3 and y = 7.
Check a candidate pair in both original equations. Satisfying one equation alone is not enough.

Worked example: Two adult and three child tickets cost £31; three adult and two child tickets cost £34. Set 2a + 3c = 31 and 3a + 2c = 34. Eliminating gives c = £5 and a = £8.
Graphical intersections give approximate solutions limited by scale. Parallel distinct lines have no solution; equations describing the same line have infinitely many pairs.
Define the variables with units before modelling, and interpret the pair in the context rather than reporting bare x and y values.

Substitute the linear expression into the quadratic equation to obtain an equation in one variable. Solve it, then recover the other coordinate for each root.
Worked example: y = x + 2 and y = x². Set x² = x + 2, giving (x − 2)(x + 1) = 0. The solutions are (2, 4) and (−1, 1).
Intersections of y equals x plus two and y equals x squared
Enlarge diagram: Intersections of y equals x plus two and y equals x squared
For x² + y² = 25 and y = x + 1, substitute the whole bracket: x² + (x + 1)² = 25. Expand before solving; (x + 1)² is not x² + 1.
A line can meet a parabola or circle twice, once or not at all. Keep corresponding x and y values paired; do not mix coordinates from different solutions.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.