Solving Systems by Elimination Practice Test

Solve linear systems by adding or subtracting scaled equations.

Solving Systems by Elimination Practice Test

This test has 20 questions

After the test · coefficient canceller

Elimination solves a system by combining the two equations so one variable disappears

The goal is to create opposite coefficients for one variable, then add the equations so those terms cancel. Sometimes the system is already prepared for elimination. Other times you multiply one or both equations first. After one variable is found, use either original equation to recover the second coordinate and verify the ordered pair.

Opposite coefficientsAdd equationsSubtract equations Scale one rowScale both rowsBack-solveVerify pair
Multiplying an equation by a nonzero number changes its appearance but not its solution set. That is what makes coefficient matching valid.
System
2x + 3y = 13
x − 3y = 2
Opposite terms
+3y and −3y
Combine
(2x + 3y) + (x − 3y) = 13 + 2
Cancel
3y + (−3y) = 0
Result
3x = 15

Elimination readiness scan

Before changing either equation, check whether a variable is already ready to cancel

Already opposite
2x + 3y = 13
x − 3y = 2
Add immediately: +3y and −3y cancel.
Same coefficients
3x + 2y = 14
x + 2y = 8
Subtract the equations so +2y − +2y = 0.
One easy multiple
2x + y = 7
3x − 2y = 7
Multiply the first equation by 2 to create +2y and −2y.
Both need scaling
2x + 3y = 12
5x + 2y = 16
Use multiples that create opposite x- or y-coefficients.
Full elimination grid · 2x + 3y = 13 and x − 3y = 2
ALIGN
2x + 3y = 13
x − 3y = 2 Write matching variable columns under each other.
ADD
3x + 0y = 15 Add left sides and right sides. The y-terms cancel.
SOLVE x
3x = 15 → x = 5 The system is now reduced to one variable.
BACK
5 − 3y = 2 → −3y = −3 → y = 1 Use either original equation to find the second coordinate.
PAIR
(5, 1) Write x first and y second.

Scale one equation

If one coefficient is an easy multiple of the other, change only the row that needs it

Before scaling

2x + y = 7
3x − 2y = 7

The y-coefficients +1 and −2 are not opposites yet.

×2
FIRST
ROW

Ready to eliminate

4x + 2y = 14
3x − 2y = 7

7x = 21 → x = 3

Now +2y and −2y cancel by addition. Back-substitution gives y = 1.

Scale both equations

When neither coefficient is an easy multiple of the other, use a common multiple

Original
2x + 3y = 12
5x + 2y = 16
For y, a convenient common coefficient is 6.
First row ×2
4x + 6y = 24
The y-coefficient becomes +6.
Second row ×−3
−15x − 6y = −48
The y-coefficient becomes −6.
Add
−11x = −24 → x = 24/11
A fractional result is valid; elimination does not guarantee integer coordinates.

Add or subtract?

Choose the row operation that actually makes one coefficient pair disappear

Add when coefficients are opposites

+4x and −4x
+3y and −3y

Opposite numbers sum to zero, so ordinary row addition performs the cancellation.

Subtract when coefficients are equal

+5x and +5x
−2y and −2y

Subtracting one entire equation from the other creates zero in that variable column.

Known value
x = 3
Original equation
2x + y = 7
Substitute x
2(3) + y = 7
Solve y
6 + y = 7 → y = 1
Ordered pair
(3, 1)

Two-row verification

Check the final ordered pair in both original equations

Pair
Equation 1
Equation 2
Status
(5, 1)
2(5) + 3(1) = 13
5 − 3(1) = 2
both true ✓

Elimination outcome detector

Sometimes both variables disappear; the remaining statement classifies the system

This is a feature of the method, not a dead end.

Variable remains
3x = 15 → x = 5
one solution
Contradiction
0x + 0y = 7 → 0 = 7
no solution
Identity
0x + 0y = 0 → 0 = 0
infinitely many

Elimination word model

Elimination is useful when two relationships are already written with aligned coefficients

Adult tickets
x
Student tickets
y
Total sold
x + y = 50
Total revenue
12x + 8y = 520
First equation ×−8
−8x − 8y = −400
Add to revenue equation
4x = 120
Solution
x = 30, y = 20

Elimination error scan

Most errors come from scaling only part of an equation, misaligning terms, or combining signs incorrectly

Partial row scaling
2x + y = 7 multiplied by 2 → 4x + y = 14 4x + 2y = 14

Every term on both sides of the equation must be multiplied.

Like columns misaligned
x-term added to a y-term because the equations were written out of alignment. Keep x, y, and constants in separate vertical columns.

Only like terms may be combined.

Wrong sign in addition
3y + (−3y) = 6y 3y + (−3y) = 0

Opposite coefficients cancel; they do not double.

Stopped after one variable
x = 5 reported as the complete answer. Back-solve for y and report the ordered pair.

A two-variable system normally requires both coordinates.

Coefficient-canceller diagnostics

Sort missed questions by the exact elimination step that failed

Coefficient target Was the best variable chosen for cancellation?
Row scaling Were all terms multiplied correctly to create matching or opposite coefficients?
Add / subtract Was the correct row operation used and were signed terms combined accurately?
Back-solve & verify Was the second variable found and the ordered pair checked in both originals?