Sheet № 23 · Foundation + Higher · AQA · Edexcel · OCR
Simultaneous Equations: Elimination –
Simultaneous equations involve two equations with two unknowns, and you need to find values that satisfy both equations at the same time. The elimination method works by adding or subtracting the equations to remove one variable, leaving a single equation in one unknown that you can solve directly. This topic appears on both Foundation an
§Key definitions
Question:
Solve simultaneously: 3x + 2y = 16 and x + 2y = 10.
Check using ①:
3(3) + 2(3.5) = 9 + 7 = 16 ✓
Answer:
x = 3, y = 3.5
Check using ②:
5(4) − 2(3.5) = 20 − 7 = 13 ✓
Q1 (Foundation):
Solve: 2x + y = 11 and 2x − y = 5.
§Formulas to memorise
To eliminate a variable, make the coefficients of that variable the same in both equations, then add or subtract.
If the matching coefficients have the same sign, subtract the equations. If they have opposite signs, add them.
Label the equations — ① and ② for easy reference.
Choose which variable to eliminate. — Pick the one whose coefficients are easiest to match.
Multiply one or both equations — so that the chosen variable has the same coefficient in both.
Add or subtract — the equations to eliminate that variable.
Solve — the resulting single equation.
Substitute — the value you found back into one of the original equations to find the other variable.
Check — by substituting both values into the other original equation.
Worked example
Solve simultaneously: 3x + 2y = 16 and x + 2y = 10.
Working:
⚠ Common mistakes
- ✗Subtracting when you should add (or vice versa). Use the SSSD rule: same signs subtract, different signs add. Write the signs explicitly before deciding.
- ✗Forgetting to multiply every term. When you multiply an equation by 3, every term — including the constant on the right — must be multiplied.
- ✗Substituting back into the wrong equation. It does not matter which you use, but always check using the other equation to verify your answer.
- ✗Arithmetic errors in the addition/subtraction step. Write the equations directly above each other and line up the terms. This makes addition and subtraction much clearer.
- ✗Giving only one value. You must state both x and y. A single value does not fully solve the system.
✦ Exam tips
- →Label your equations ① and ② and refer to them by number throughout. Examiners appreciate structured working.
- →Show the multiplication step clearly. Write "①×2:" before the new equation. This earns explicit method marks.
- →Always check in the equation you did not use for substitution. If both values satisfy both equations, you can be confident in your answer.
- →If the answer is a fraction or decimal, that is fine. Not every simultaneous equation has neat whole-number answers, especially on Higher papers.