Linear equations in two unknowns
Solve simultaneous linear equations by substitution and elimination — core Secondary 3 skills.
Key formulas
What a pair of equations means
Two linear equations in x and y usually determine a unique intersection point. Graphically they are two straight lines; algebraically we seek the common (x, y).
Substitution method
Make one variable the subject of the simpler equation, substitute into the other, solve, then back-substitute. Ideal when a coefficient is already 1 or −1.
Elimination method
Multiply equations so that one variable has equal (or opposite) coefficients, then add or subtract to eliminate it. Keep a clear working column for each equation.
Exam habit
State the method. Box the final pair (x, y) and check both original equations.
Worked example
Elimination
Problem. Solve 2x + y = 8 and x − y = 1.
Solution. Add: 3x = 9 ⇒ x = 3. Then 3 − y = 1 ⇒ y = 2. Check both equations.
Review quiz
1 / 4x + y = 10, x − y = 2. Then x =
Practice questions
- Solve: x + y = 10 and x − y = 2.
- Solve: 2x + y = 7 and x − y = 2 by elimination.
- Solve: y = 2x − 1 and 3x + 2y = 12 by substitution.
Study notes
When is there no solution?
Parallel lines — the elimination step produces a false statement such as 0 = 4.
Infinite solutions?
The equations are multiples of each other; elimination produces 0 = 0.