Systems of Linear Equations • Topic 3 of 4

Elimination Method

Elimination adds or subtracts the two equations to cancel one variable. If the coefficients of a variable are equal and opposite, adding eliminates it; if they are equal, subtracting does. Sometimes you first multiply an equation by a constant so a variable lines up. For x + y = 10 and x − y = 4, adding gives 2x = 14, so x = 7 and y = 3. Elimination is efficient when no variable is isolated and coefficients match or can be matched easily — a frequent SAT setup with "ax + by = c" form equations.

✅ Solved examples

1. Solve x + y = 10 and x − y = 4.
Add the equations: 2x = 14 → x = 7; then y = 3.
2. Solve 2x + y = 11 and x − y = 1 by elimination.
Add: 3x = 12 → x = 4; then y = 3.
3. Solve 2x + y = 7 and 2x − y = 1.
Add the equations: 4x = 8, so x = 2. Substitute back: 2(2) + y = 7, so y = 3. Solution (2, 3).
4. Solve x + 2y = 8 and x + y = 5.
Subtract: y = 3; then x = 2.

✏️ Practice — try these, take hints as needed

1. Solve x + y = 12 and x − y = 2.
Add the equations to cancel y.
2x = 14.
Find x then y.
x = 7, y = 5.
2. Solve 2x + y = 9 and x − y = 0.
Add to cancel y.
3x = 9.
Find x then y.
x = 3, y = 3.
3. Solve x + 3y = 14 and x + y = 6.
Subtract the equations.
2y = 8.
Find y then x.
y = 4, x = 2.
4. Solve 3x + y = 10 and x + y = 4.
Subtract to cancel y.
2x = 6.
Find x then y.
x = 3, y = 1.
5. Solve x + 2y = 11 and x − 2y = 3.
Add the equations to cancel y.
2x = 14.
Find x, then substitute for y.
x = 7, y = 2.

📝 Topic test — 8 questions

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