Simultaneous linear equations · Ages 13–14
From words to simultaneous equations
Two unknowns need two different facts.
How to do it
Method
Two unknowns need two different facts
2x + 3y = 260, x − y = 30
Another way
Just one unknown
2(y + 30) + 3y = 260
In full
Two unknown quantities: choose two unknowns, say $x for an adult ticket and $y for a child's. Then find two different facts and write each as an equation: "2 adult and 3 child tickets cost $260" is 2x + 3y = 260; "an adult ticket costs $30 more" is x − y = 30. Each fact is used once, and the two equations must say different things. Or use one unknown: a child's ticket $y, an adult's $(y + 30), and 2(y + 30) + 3y = 260, the same answer. Then try real values to see that the equations fit the story.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Two numbers add to 74 and differ by 6. With x the larger and y the smaller: x + y = , x − y = , so x = .
Fill in each blank
Show the answer and working
Answer: 74, 6, 40
Sum: x + y = 74; difference: x − y = 6. Adding, 2x = 74 + 6, x = 40.
Example 2
Chickens and rabbits: x + y = 14, 2x + 4y = 48 (x chickens, y rabbits). Double the first and take it from the second: y =
Show the answer and working
Answer: 10
2x + 2y = 28; (2x + 4y) − (2x + 2y) = 48 − 28, so 2y = 20 and y = 10.
Example 3
A rectangle's perimeter is 38 cm, and its length is 2 cm less than 2 times its width. With length x cm and width y cm, which pair of equations is right?
- A. x + y = 38, x = 2y − 2
- B. 2x + 2y = 38, y = 2x − 2
- C. 2x + 2y = 38, x = 2y − 2
- D. 2x + 2y = 38, x = 2(y − 2)
Choose the right answer
Show the answer and working
Answer: C. 2x + 2y = 38, x = 2y − 2
Perimeter: 2x + 2y = 38; length from width: x = 2y − 2. Try x = 12, y = 7: both hold.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.An adult ticket is $x and a child's $y. 1 adult and 3 child tickets cost $290; 1 adult and 2 child tickets cost $220. Which pair of equations is right?
- A. 3x + y = 290, 2x + y = 220
- B. x + 3y = 290, x + 2y = 220
- C. x + 3y = 220, x + 2y = 290
- D. x + 3y = 290, 2x + y = 220
Choose the right answer
Answer:
2.An apple is $x and an orange $y. 5 apples and 2 oranges cost $99, and an apple costs $3 more than an orange. Which pair of equations is right?
- A. 5x + 2y = 99, x − y = 3
- B. 5x + 2y = 99, x + y = 3
- C. 2x + 5y = 99, x − y = 3
- D. 5x + 2y = 99, y − x = 3
Choose the right answer
Answer:
3.Two numbers add to 66 and differ by 6. With x the larger and y the smaller: x + y = , x − y = , so x = .
Fill in each blank
Answer:
4.If Chi-keung gives Tsz-hin $4, they have the same; if Tsz-hin gives Chi-keung $4, Chi-keung has 2 times as much as Tsz-hin. With $x as Chi-keung's and $y as Tsz-hin's, which pair of equations is right?
- A. x − 4 = y + 4, x + 4 = 2(y − 4)
- B. x − 4 = y + 4, 2(x + 4) = y − 4
- C. x − 4 = y, x + 4 = 2(y − 4)
- D. x − 4 = y + 4, x + 4 = 2y − 4
Choose the right answer
Answer:
5.Chickens and rabbits: x + y = 26, 2x + 4y = 82 (x chickens, y rabbits). Double the first and take it from the second: y =
Answer:
All the answers
- 1. B. x + 3y = 290, x + 2y = 220
- 2. A. 5x + 2y = 99, x − y = 3
- 3. 66, 6, 36
- 4. A. x − 4 = y + 4, x + 4 = 2(y − 4)
- 5. 15
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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