Simultaneous linear equations · Ages 13–14
Solving by graphs
Where the lines cross is the solution; check it in both.
How to do it
Method
Where the lines cross is the solution; check it in both
x + y = 5 and x − y = 1 meet at (3, 2)
Another way
Check by substituting
3 + 2 = 5, 3 − 2 = 1
In full
A pair of simultaneous equations asks for one (x, y) that makes both true. Each equation's graph is a straight line, and the point where they cross is on both, so its coordinates are the solution. Two points draw a line, and x = 0 and y = 0 are the easiest: x + y = 5 passes through (0, 5) and (5, 0). x + y = 5 and x − y = 1 meet at (3, 2): x = 3, y = 2. Once read, put the point back into both to check: 3 + 2 = 5 and 3 − 2 = 1. Parallel lines never meet, so there is no solution; lines lying on top of each other give endlessly many.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Solve ① x + 2y = 4 and ② 3x − y = −16 by the graphs: x = , y =
Fill in each blank
Show the answer and working
Answer: −4, 4
The lines meet at (−4, 4). Check: ①'s left side is 4 and ②'s is −16.
Example 2
The picture shows lines ① and ②. Reading it, what is the solution (x, y) of the pair of equations?
- A. (2, −3)
- B. (−3, 2)
- C. (−3, −2)
- D. (3, 2)
Choose the right answer
Show the answer and working
Answer: B. (−3, 2)
Where the lines cross is on both, so its coordinates, (−3, 2), are the solution.
Example 3
To draw x + 3y = 6, find two points first: (0, ) and (, 0).
Fill in each blank
Show the answer and working
Answer: 2, 6
With x = 0, 3y = 6, y = 2; with y = 0, x = 6, x = 6. Join the two points.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The picture shows lines ① and ②. Reading it, what is the solution (x, y) of the pair of equations?
- A. (1, −2)
- B. (−1, 2)
- C. (1, 2)
- D. (2, 1)
Choose the right answer
Answer:
2.Which pair of equations has graphs that are two parallel lines (no solution)?
- A. 4x + 2y = 3 and 12x + 6y = 12
- B. 4x + 2y = 3 and 12x + 6y = 9
- C. 4x + 2y = 3 and 4x − 2y = 3
- D. 4x + 2y = 3 and 2x + 4y = 3
Choose the right answer
Answer:
3.Lines ① and ② cross in the picture. With the crossing at (x, y), x + y =
Answer:
4.The picture shows that x = −3, y = −4 satisfies both ① and ②.
True or false?
Answer:
5.To draw 4x + y = 4, find two points first: (0, ) and (, 0).
Fill in each blank
Answer:
All the answers
- 1. C. (1, 2)
- 2. A. 4x + 2y = 3 and 12x + 6y = 12
- 3. 3
- 4. True
- 5. 4, 1
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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