Systems of Equations • Topic 2 of 4

Elimination

Elimination adds or subtracts the two equations so that one variable disappears. If a variable already has matching coefficients — like 2x in both — subtract the equations to cancel it; if the coefficients are opposites, add. When neither lines up, multiply one or both equations by whatever numbers make one variable's coefficients equal in size, then combine. Solving x + y = 10 and x − y = 4 is instant: add them to get 2x = 14, x = 7, then y = 3. Elimination is usually faster than substitution when no coefficient is 1, and it is the natural choice on the GRE when a question asks for x + y or x − y directly — you may get the answer from a single addition or subtraction without ever finding x and y separately.

✅ Solved examples

1. Solve x + y = 10 and x − y = 4.
Add the equations: 2x = 14 ⇒ x = 7. Subtract to check: 2y = 6 ⇒ y = 3. Solution (7, 3).
2. Solve 2x + 3y = 16 and 2x − y = 4.
Subtract: (2x + 3y) − (2x − y) = 16 − 4 ⇒ 4y = 12 ⇒ y = 3. Then 2x − 3 = 4 ⇒ x = 3.5. Solution (3.5, 3).
3. Solve 3x + 2y = 12 and x + 2y = 8.
Subtract: 2x = 4 ⇒ x = 2. Then 2 + 2y = 8 ⇒ y = 3. Solution (2, 3).
4. Solve 2x + 3y = 13 and 4x − 3y = 5. Find x + y.
Add: 6x = 18 ⇒ x = 3. Then 6 + 3y = 13 ⇒ y = 7/3. So x + y = 3 + 7/3 = 16/3.

✏️ Practice — try these, take hints as needed

1. Solve x + y = 12 and x − y = 2.
Add the two equations.
2x = 14.
Then find y.
x = 7, y = 5
2. Solve 3x + y = 14 and 3x − y = 4.
Add to cancel y.
6x = 18.
x = 3, then solve y.
x = 3, y = 5
3. Solve 5x + 2y = 20 and 5x − 2y = 10.
Add to cancel y.
10x = 30.
x = 3.
x = 3, y = 2.5
4. Solve 2x + 3y = 18 and 2x + y = 10.
Subtract the equations.
2y = 8.
y = 4, then x.
x = 3, y = 4
5. Solve x + 2y = 9 and 3x − 2y = 7. Find x.
Add to cancel y.
4x = 16.
x = 4.
x = 4, y = 2.5

📝 Topic test — 8 questions

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