Simultaneous linear equations · Ages 13–14
Solutions of an equation in two unknowns
A pair that balances it is a solution; they all lie on one line.
How to do it
Method
A pair that balances it is a solution; they all lie on one line
(6, 1): 6 + 2 × 1 = 8, a solution
Another way
y in terms of x, then a table
y = 8 − x2: x = 6, y = 1
In full
x + 2y = 8 has two unknowns, each to the first power: a linear equation in two unknowns. Every pair (x, y) that makes it true is a solution, and there are endlessly many: (0, 4), (2, 3), (4, 2), (8, 0), … To test (6, 1), put in x = 6 and y = 1: 6 + 2 × 1 = 8, both sides match, so it is one. Plotted on the coordinate plane, the solutions all lie on one straight line: that line is the equation's graph. Or write y in terms of x first, y = 8 − x2, and make a table one x at a time.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Which point lies on the line x + 6y = −19?
- A. (5, −5)
- B. (6, −4)
- C. (5, −4)
- D. (−4, 5)
Choose the right answer
Show the answer and working
Answer: C. (5, −4)
A point is on the line when its coordinates solve the equation: 1 × (5) + 6 × (−4) = −19.
Example 2
6x + 5y = 90: when x = 0, y = ; when y = 0, x = .
Fill in each blank
Show the answer and working
Answer: 18, 15
x = 0: 5y = 90, y = 18; y = 0: 6x = 90, x = 15. These two points draw the line most easily.
Example 3
Write x + 3y = 21 as y = − x3. When x = 6, y = ; when x = 9, y = .
Fill in each blank
Show the answer and working
Answer: 21, 5, 4
Take x from both sides, then divide by 3: y = 21 − x3. x = 6: 21 − 63 = 5; x = 9: 21 − 93 = 4. Each time x goes up by 3, y goes down by 1.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In x + 2y = −7, when x = 1, y =
Answer:
2.In x − y = 0, when y = 4, x =
Answer:
3.x + 6y = 12: when x = 0, y = ; when y = 0, x = .
Fill in each blank
Answer:
4.Write 6x + 5y = 69 as y = − 6x5. When x = 4, y = ; when x = 9, y = .
Fill in each blank
Answer:
5.Which of these points is not on the line 5x + 4y = 11?
- A. (−5, 9)
- B. (−1, 4)
- C. (3, −1)
- D. (3, 9)
Choose the right answer
Answer:
All the answers
- 1. −4
- 2. 4
- 3. 2, 12
- 4. 69, 9, 3
- 5. D. (3, 9)
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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