Simultaneous Equations — Elimination
Grade 5Why two equations?
One equation with two unknowns has infinitely many solutions. Two equations with the same two unknowns usually pins down one solution. Simultaneous equations are a pair of equations you solve together.
The elimination method works by adding or subtracting the equations to remove one variable. Then you solve for the remaining one and substitute back.
Worked example — same coefficient already
EXAMPLE
Solve simultaneously: 3x + 2y = 16 and x + 2y = 8.
The coefficient of y is 2 in both equations. Subtract the second from the first: (3x + 2y) − (x + 2y) = 16 − 8, giving 2x = 8, so x = 4. Substitute into x + 2y = 8: 4 + 2y = 8, so 2y = 4, y = 2. Answer: x = 4, y = 2. Check in eq 1: 3(4) + 2(2) = 16 ✓.
When coefficients do not match — scaling up
If no variable has matching coefficients, multiply one (or both) equations by a suitable number to make a coefficient match, then eliminate.
EXAMPLE
Solve: 2x + 3y = 13 and 5x − y = 7.
Multiply the second equation by 3: 15x − 3y = 21. Now add to the first: 2x + 3y + 15x − 3y = 13 + 21, giving 17x = 34, so x = 2. Substitute into 5x − y = 7: 10 − y = 7, so y = 3. Answer: x = 2, y = 3.
Always verify both equations
Substitute your answers into the equation you did not use to find y. If it works, you are done. If not, go back and check your arithmetic.
Try this one
Solve the simultaneous equations.
y = 2x + 1
y = x + 5
[3 marks]
x = __________ y = __________
