Sheet № 24 · Foundation + Higher · AQA · Edexcel · OCR
Simultaneous Equations: Substitution –
The substitution method is the second key technique for solving simultaneous equations, and it truly comes into its own when one equation is already solved for a single variable — or when you have a linear and a non-linear equation paired together. On Foundation papers you will meet purely linear pairs, while Higher tier extends this to s
§Key definitions
Question:
Solve simultaneously: y = 3x − 1 and 2x + y = 9.
Check using 2x + y = 9:
2(2) + 5 = 9 ✓
Answer:
x = 2, y = 5
Check for x = −2, y = −3:
(−2)² + (−3)² = 4 + 9 = 13 ✓
Q1 (Foundation):
Solve: y = x + 4 and 3x + 2y = 23.
§Formulas to memorise
If y = mx + c and you have a second equation, replace every y in the second equation with (mx + c).
For linear + quadratic systems, substitution typically produces a quadratic equation to solve by factorising or using the quadratic formula.
One equation is already in the form y = ... or x = ...
Pick the equation where a variable is easiest to isolate. — For example, if one equation is x + 2y = 10, rearrange to x = 10 − 2y.
Substitute — this expression into the other equation. Replace every x with (10 − 2y).
Solve — the resulting equation for y.
Substitute — the value of y back into the rearranged equation to find x.
Check — both values in the equation you did not use for substitution.
Rearrange the linear equation — to make y (or x) the subject.
Substitute — into the non-linear equation (e.g., x² + y² = 25 or y = x² − 3x).
Worked example
Solve simultaneously: y = 3x − 1 and 2x + y = 9.
Working:
⚠ Common mistakes
- ✗Substituting into the same equation you rearranged. This leads to a tautology like y = y, which tells you nothing. Always substitute into the other equation.
- ✗Forgetting to square the entire expression. When substituting y = 2x + 1 into y², you must expand (2x + 1)², not write 2x² + 1. The cross term (4x) is essential.
- ✗Missing one solution in non-linear systems. A linear-quadratic system typically has two solutions. If you find only one, check your factorising or quadratic formula work.
- ✗Not pairing x and y values correctly. Each x value has a specific y value that goes with it. Write your solutions as ordered pairs to avoid confusion.
- ✗Arithmetic errors during expansion. Write out the expansion of (2x + 1)² fully, term by term: (2x)(2x) + (2x)(1) + (1)(2x) + (1)(1) = 4x² + 4x + 1.
✦ Exam tips
- →If one equation is already y = ... or x = ..., use substitution. It is faster than elimination in this case.
- →For linear + non-linear systems, substitution is the only option. Elimination cannot remove a variable from an x² or y² term. This is a Higher-only question type.
- →Present both solutions clearly as pairs. Write: "x = 2, y = 5" and "x = −1, y = 3" on separate lines or as coordinate pairs.
- →If your quadratic does not factorise, use the quadratic formula and give answers to an appropriate degree of accuracy. See Solving Quadratic Equations — Quadratic Formula for guidance.