4.4Solving Systems by Elimination
Add or subtract equations to eliminate one variable, then solve for the other — the fastest algebraic method when coefficients are opposites or equal.
Why This Matters
Elimination is the most efficient method for many systems and is the basis for Gaussian elimination used in linear algebra and computer science. It's also the fastest approach on timed tests like the SAT and ACT.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does adding or subtracting two equations eliminate a variable, and when is the elimination method more efficient than graphing or substitution?
Lesson Overview
The elimination method (also called the addition method) solves a system by adding or subtracting the two equations so that one variable is cancelled out. This leaves a single equation with one unknown, which is easy to solve. Elimination is most efficient when the coefficients of one variable are already opposites (e.g., +3y and −3y) or equal (e.g., 2y and 2y). When neither condition holds, we multiply one or both equations by a constant to create opposite coefficients before adding.
5-step elimination method
Before → After Elimination
Before (2 equations, 2 unknowns)
x + y = 7
x − y = 1
After (1 equation, 1 unknown)
2x = 8
x = 4
y = 3
Adding eliminates y — leaving only x
Elimination reduces two equations to one
Elimination: add equations → x+y=7 and x−y=1 → (4,3)
Adding Opposite Coefficients
x + y = 7
+ x − y = 1
2x = 8 → x = 4
y-terms cancel → one variable left!
Adding equations to eliminate y
6x = 18 → x = 3
y terms cancelled — one variable left!
Opposite coefficients cancel the y-terms
When to add vs. subtract
Choosing the best method
Graphing · Substitution · Elimination compared
Which method should I use?
Is one variable already isolated?
Yes → Substitution
Do coefficients of one variable sum to zero (or match)?
Yes → Elimination
Are both equations in slope-intercept form?
Yes → Graphing
Multiply to create opposite coefficients → Elimination
Decision tree: choosing the best method
Worked Examples
Step 1 — Align
2x + 3y = 12
4x − 3y = 6
Standard form, aligned
Step 2 — Add
6x + 0y = 18
y-terms cancel (±3y)
Step 3 — Solve
x = 3, then y = 2
Back-substitute & verify
Solve by elimination: x + y = 7 and x − y = 1
Both equations are in standard form.
The y-coefficients are +1 and −1 — they are opposites. Add the equations.
(x + y) + (x − y) = 7 + 1 → 2x = 8 → x = 4.
Back-substitute into Equation 1: 4 + y = 7 → y = 3.
Verify: Eq 1: 4+3=7 ✓ Eq 2: 4−3=1 ✓
Graph confirms solution (4, 3)
Verify (4, 3)
Equation 1
4+3=7
7=7 ✓
Equation 2
4−3=1
1=1 ✓
Both satisfied ✓
Solve by elimination: 2x + 3y = 12 and 4x − 3y = 6
The y-coefficients are +3 and −3 — opposites. Add the equations.
(2x + 3y) + (4x − 3y) = 12 + 6 → 6x = 18 → x = 3.
Back-substitute into Equation 1: 2(3) + 3y = 12 → 6 + 3y = 12 → 3y = 6 → y = 2.
Verify: Eq 1: 2(3)+3(2)=6+6=12 ✓ Eq 2: 4(3)−3(2)=12−6=6 ✓
Graph confirms solution (3, 2)
Solve by elimination: 3x + 2y = 16 and x + 2y = 8
The y-coefficients are both +2 — equal. Subtract Equation 2 from Equation 1.
(3x + 2y) − (x + 2y) = 16 − 8 → 2x = 8 → x = 4.
Back-substitute into Equation 2: 4 + 2y = 8 → 2y = 4 → y = 2.
Verify: Eq 1: 3(4)+2(2)=12+4=16 ✓ Eq 2: 4+2(2)=8 ✓
Graph confirms solution (4, 2)
Solve by elimination: 2x + y = 9 and x − y = 3
The y-coefficients are +1 and −1 — opposites. Add the equations.
(2x + y) + (x − y) = 9 + 3 → 3x = 12 → x = 4.
Back-substitute into Equation 2: 4 − y = 3 → y = 1.
Verify: Eq 1: 2(4)+1=9 ✓ Eq 2: 4−1=3 ✓
Graph confirms solution (4, 1)
Solve by elimination: 2x + y = 5 and 2x + y = 8
Subtract Equation 1 from Equation 2: (2x+y) − (2x+y) = 8 − 5.
0 = 3 — this is a contradiction.
The system has no solution — the lines are parallel.
No Solution — Inconsistent System
Guided Practice
Guided Practice Video: Solving Systems by Elimination
Watch the guided practice walkthrough for solving systems by elimination, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve by elimination: 3x + y = 10 and x − y = 2.
Hint: The y-coefficients are +1 and −1 — they are opposites. Add the equations to eliminate y.
Solve by elimination: 5x + 2y = 14 and 3x + 2y = 10.
Hint: The y-coefficients are both +2 — they are equal. Subtract the second equation from the first to eliminate y.
Solve by elimination: 2x + 3y = 12 and x − y = 1.
Hint: To eliminate y, multiply Equation 2 by 3 to get 3x − 3y = 3. Now the y-coefficients are +3 and −3.
Solve by elimination: 4x − 3y = 1 and 2x + 3y = 11.
Hint: The y-coefficients are −3 and +3 — opposites. Add the equations directly.
Solve by elimination: 3x + 2y = 7 and 6x + 4y = 10. How many solutions?
Hint: Multiply Equation 1 by 2 to get 6x + 4y = 14. Now subtract from Equation 2. What do you get?
Key Vocabulary
Elimination Method
A technique for solving systems by adding or subtracting equations to cancel one variable.
Opposite Coefficients
Coefficients that are equal in magnitude but opposite in sign (e.g., +3 and −3). Adding equations with opposite coefficients eliminates that variable.
Equal Coefficients
Coefficients that are the same value. Subtracting equations with equal coefficients eliminates that variable.
Multiplier
A constant used to scale an equation so that a variable's coefficient becomes opposite to the corresponding coefficient in the other equation.
Standard Form
Ax + By = C — the form equations should be in before applying elimination.
Contradiction
A false statement (e.g., 0 = 5) produced by elimination, indicating the system has no solution (parallel lines).
Identity
A statement that is always true (e.g., 0 = 0) produced by elimination, indicating infinitely many solutions (same line).
Back-Substitution
After finding one variable's value, substituting it into either original equation to find the other variable.
Interactive Practice — 5 Questions
Solve by elimination: x + y = 10 and x − y = 4.
When solving by elimination, you arrive at 0 = 7. What does this mean?
To solve 2x + 3y = 11 and x − y = 2 by elimination, what is the best first step?
Which system is best solved by elimination?
Solve by elimination: 3x + 2y = 16 and 3x − 2y = 8.
Independent Practice
Independent Practice
Solve by elimination: x + y = 9 and x − y = 3.
Solve by elimination: 2x + y = 11 and 2x − y = 5.
Solve by elimination: 3x + 2y = 14 and x + 2y = 6.
Solve by elimination: 4x − y = 7 and 2x + y = 5.
Solve by elimination: 5x + 3y = 17 and 5x − 3y = 7.
Common Mistakes
Multiplying only the left side of an equation when scaling — e.g., ×3: 9x + 6y = 12 (right side not multiplied).
Multiply every term on both sides: 3x + 2y = 4 → ×3 → 9x + 6y = 12.
Adding equations when you should subtract (or vice versa) — ending up with a term that doesn't cancel.
Check the signs of the matching coefficients. If they're opposite, add. If they're the same, subtract.
Forgetting to find the second variable after eliminating the first.
After finding one variable, substitute back into either original equation to find the other. Write (x, y).
Choosing multipliers that create equal coefficients but forgetting to make them opposite in sign for addition.
To eliminate by adding, the coefficients must be opposites (e.g., +6y and −6y). Scale one or both equations accordingly.
Math Tips
Opposite coefficients (e.g., +3y and −3y) → add the equations to cancel that variable.
Equal coefficients (e.g., both +2y) → subtract one equation from the other to cancel.
When multiplying to create opposite coefficients, multiply every term — including the constant on the right side.
A contradiction (0 = k, k ≠ 0) means no solution; an identity (0 = 0) means infinitely many solutions.
Always verify your final (x, y) by substituting into both original equations.