Solving Systems by Elimination
Key moves, shortcuts, and special cases for eliminating a variable on PSAT systems.
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Questions Covered in This Set
11 cards to master
What is the core idea of elimination?
If two equations are both true, adding or subtracting them keeps a true equation — choose the combination so one variable cancels.
When do you ADD the two equations directly?
When a variable's coefficients are opposite in sign and equal in size (e.g. +2y and -2y).
When do you SUBTRACT the equations?
When a variable's coefficients are identical (same sign, same size), like 4x and 4x.
Most common student error when subtracting equations?
Forgetting to distribute the minus sign to EVERY term of the second equation.
Solve: 3x + 2y = 16 and 5x - 2y = 8
Add: 8x = 24, x = 3; then 9 + 2y = 16, y = 3.5. Solution (3, 3.5).
What do you do when no coefficients match?
Scale: multiply a whole equation by a constant to create matching (or opposite) coefficients, using an LCM.
How do you choose which variable to eliminate?
Pick the variable whose coefficients need the smaller numbers (smaller LCM) — or that already cancels.
If 4x + 5y = 22 and 5x + 4y = 23, find x + y
Add: 9x + 9y = 45, so x + y = 5 — no need to find x and y separately.
Elimination leaves 0 = 0. What does that mean?
Infinitely many solutions — the equations describe the same line.
Elimination leaves 0 = 7. What does that mean?
No solution — the lines are parallel.
For what c do 3x + 6y = 9 and x + cy = 3 have infinitely many solutions?
Multiply the second by 3: 3x + 3cy = 9, so 3c = 6 and c = 2.
