Form a pair of linear equations and find their common solutions graphically.
10 students took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
Answer: x + y = 10 and x − y = 4 (x = girls, y = boys). The lines meet at (7, 3): 7 girls and 3 boys.
Step-by-step solution
Idea: Draw both lines from a small table of values and read off the point where they cross.
- Let the number of girls be x and boys y. Total: x + y = 10. Girls are 4 more: x − y = 4.1 mark
- Tables:
x + y = 10:
x − y = 4:x 0 5 10 y 10 5 0
1 markx 4 6 7 y 0 2 3 - Plot the points and draw the two lines (see the graph). They intersect at (7, 3).1 mark
- So x = 7, y = 3: 7 girls and 3 boys took part.1 mark
7 girls and 3 boys (the lines x + y = 10 and x − y = 4 meet at (7, 3)).
Check: 7 + 3 = 10 ✓ and 7 − 3 = 4 ✓.
Answer to write in the exam
Let girls = x, boys = y
x + y = 10 … (1); x − y = 4 … (2)
(1):
| x | 0 | 5 | 10 |
| y | 10 | 5 | 0 |
(2):
| x | 4 | 6 | 7 |
| y | 0 | 2 | 3 |
The lines intersect at (7, 3)
∴ Girls = 7, boys = 3
Common mistakes that cost marks
- Writing y = x + 4 (boys 4 more than girls), which gives 3 girls and 7 boys.
- Reading the intersection from a rough sketch as (6, 4); always check the point in both equations.
How this can come in the exam
Short answer (3 marks)
Graphically find two numbers whose sum is 8 and whose difference is 2.
Show answer
x + y = 8: (0, 8), (8, 0); x − y = 2: (2, 0), (5, 3) (1 mark). Draw both (1 mark). They meet at (5, 3): numbers 5 and 3 (1 mark).Try one yourself
In a class of 12 students, boys are 2 fewer than girls. Find each number.
Show answer
g + b = 12, g − b = 2 ⇒ 7 girls, 5 boys.
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- Here is a problem that was posed by Mahāvīrāchārya in Gaṇita sāra saṅgraha (c. 850 CE).
The price of 9 citrons and 7 fragrant wood-apples taken together is 107; and the price of 7 citrons and 9 fragrant wood-apples taken together is 101.
O mathematician, tell me quickly the price of each citron and of each fragrant wood-apple.
(Hint: The given problem can be modelled as
9x + 7y = 107
7x + 9y = 101
Can you figure out a way of solving these equations without directly using elimination or the substitution method? What is special in this pair of equations? The x and y coefficients are interchanged in the 2 equations.
What will happen if we add the pair of equations? What will happen if we subtract the pair of equations? Can the resulting equations be solved to find the values of x and y?)
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