Simultaneous linear equations · Ages 13–14
Solving by substitution
Write one unknown with the other; bracket it as you substitute.
How to do it
Method
Write one unknown with the other; bracket it as you substitute
3x + (2x − 1) = 14, x = 3, y = 5
Another way
Elimination
(2x − y) + (3x + y) = 1 + 14, 5x = 15
In full
From one equation, write one unknown in terms of the other (say y = 2x − 1) and put it into the other equation: one unknown is left. 3x + y = 14 and y = 2x − 1: 3x + (2x − 1) = 14, 5x = 15, x = 3; back into y = 2x − 1, y = 5. Put the whole expression in a bracket, above all after a minus or a coefficient: y in 3x − 2y becomes 3x − 2(2x − 1). Choose an unknown whose coefficient is 1 to write out. Or eliminate instead: y = 2x − 1 is 2x − y = 1, and adding it to 3x + y = 14 gives 5x = 15, the same answer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
y = x + 2 and 2x + y = 14: substituting, 2x + (x + 2) = 14, so x = and y = .
Fill in each blank
Show the answer and working
Answer: 4, 6
3x + 2 = 14, x = 4; back in y = x + 2, y = 6.
Example 2
x = 3y + 7, 3x − 5y = 13: x = , y =
Fill in each blank
Show the answer and working
Answer: 1, −2
Substitute: 3(3y + 7) − 5y = 13, so y = −2; back in, x = 1.
Example 3
The solution of y = 3x − 2 and x + y = 2 is x = 1, y = 2.
True or false?
Show the answer and working
Answer: False
Substituting gives x = 1, and y = 3x − 2 gives y = 1. Check the answer in both equations.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.y = 2x − 8 and 5x + y = 13: substituting, 5x + (2x − 8) = 13, so x = and y = .
Fill in each blank
Answer:
2.y = 5x, 5x + 3y = 20, x =
Answer:
3.The simultaneous equations 3x + 4y = 65 and y = 3x + k have the solution x = 3. k =
Answer:
4.2x + 3y = 12, 3x + y = 11: x = , y =
Fill in each blank
Answer:
5.4x − 5y = −43, 5x − y = −17: x = , y =
Fill in each blank
Answer:
All the answers
- 1. 3, −2
- 2. 1
- 3. 5
- 4. 3, 2
- 5. −2, 7
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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