Simultaneous linear equations · Ages 13–14
Solving by elimination
Same coefficient, subtract; opposite, add; multiply first to match.
How to do it
Method
Same coefficient, subtract; opposite, add; multiply first to match
(2x + 3y) + (4x − 3y) = 12 + 6, 6x = 18
Another way
Substitution
x = y + 2 into 3x + 2y = 16
In full
When an unknown has the same coefficient in both equations, subtract them; opposite coefficients (+3y and −3y), add them, and that unknown is gone. 2x + 3y = 12 and 4x − 3y = 6: adding, 6x = 18, x = 3; back in, 2 × 3 + 3y = 12, y = 2. When the coefficients differ, multiply one or both equations first: 3x + 2y = 16 and x − y = 2; twice the second is 2x − 2y = 4, and adding gives 5x = 20, x = 4, y = 2. Multiply every term, the number on the right too. Or substitute: x − y = 2 gives x = y + 2, into the first, and the answer is the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Multiply both sides of 6x − y = −1 by 3: 18x − 3y =
Show the answer and working
Answer: −3
Every term is multiplied by 3, the −1 on the right too: 3 × (−1) = −3.
Example 2
① 2x + y = −9, ② 5x + y = −21. Multiplying ① by 2 and taking away ② eliminates x.
True or false?
Show the answer and working
Answer: False
① × 2 has x coefficient 2 × 2; it must equal ②'s 5 for x to go.
Example 3
3x + 2y = 16, 5x − 3y = 14, y =
Show the answer and working
Answer: 2
To eliminate x: the first times 5, the second times 3, both x coefficients 3 × 5; subtract, and y = 2.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.4x + 5y = 23, 3x − 5y = −9. Adding them: x = , so x = .
Fill in each blank
Answer:
2.2x + 2y = 2 and 4x + 2y = 10. How do you eliminate y?
- A. y can't be eliminated
- B. Add the two
- C. Subtract one from the other
- D. Double the first, then add
Choose the right answer
Answer:
3.x + 2y = −3, x + y = 0: x = , y =
Fill in each blank
Answer:
4.Multiply both sides of x − y = 25 by 3: 3x − 3y =
Answer:
5.2x + 3y = 10, 5x + 2y = 3: x = , y =
Fill in each blank
Answer:
All the answers
- 1. 7, 14, 2
- 2. C. Subtract one from the other
- 3. 3, −3
- 4. 75
- 5. −1, 4
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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