Core Concept
The Idea Behind Elimination
If one equation has +3y and the other has −3y, adding them makes the y-terms cancel — eliminating that variable. Then you solve for the remaining variable.
x + 3y = 11 ... (1)
x − 3y = 1 ... (2)
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2x = 12 (add equations)
x = 6
Then: 6 + 3y = 11 → y = 5/3... (wait, let's use (1): 6+3y=11 → 3y=5 → y=5/3)
Tip: always double-check by substituting back.
Example 1
Direct Elimination
Solve: 2x + y = 8 and 3x − y = 7.
1. Notice: +y and −y — they cancel when added
2. Add equations: (2x+y) + (3x−y) = 8+7
3. 5x = 15 → x = 3
4. Substitute x=3 into Eq.1: 2(3)+y=8 → y=2
5. Solution: (3, 2)
6. Check Eq.2: 3(3)−2 = 9−2 = 7 ✓
Example 2
Multiplying to Create Opposite Coefficients
Solve: 3x + 2y = 12 and x + y = 5.
1. No variables cancel directly. Multiply Eq.2 by −2:
2. −2(x + y) = −2(5) → −2x − 2y = −10
3. Now add to Eq.1:
4. 3x + 2y = 12
5. −2x − 2y = −10
6. ──────────────
7. x = 2
8. Sub into Eq.2: 2 + y = 5 → y = 3
9. Solution: (2, 3) ✓
Special Cases
No Solution or Infinite Solutions
Sometimes when you add the equations, both variables cancel:
No Solution
x + y = 3
x + y = 7
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0 = −4 (impossible!)
Parallel lines — never intersect.
Infinite Solutions
2x + 4y = 8
x + 2y = 4
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0 = 0 (always true!)
Same line — infinite intersections.
Watch Out
Common Mistakes
- Only multiplying part of an equation — you must multiply EVERY term
- Adding when you should subtract (or vice versa) — make sure the terms you want to cancel are opposites
- Forgetting to find the second variable after eliminating the first
- Multiply an entire equation by a constant to create matching (opposite) coefficients
Your Turn
Try It Yourself
Q: Solve using elimination: 4x + 3y = 25 and 2x − 3y = 5.
Show Answer
+3y and −3y cancel when added:
6x = 30 → x = 5
4(5) + 3y = 25 → 3y = 5 → y = 5/3
Solution: (5, 5/3)
Key Takeaways
1-Minute Summary
- Create opposite coefficients by multiplying an equation by a constant
- Add the equations to eliminate one variable
- Solve for the remaining variable, then substitute back
- 0 = constant (non-zero) → no solution; 0 = 0 → infinite solutions