Systems of Sim ultaneous Equations Word problems involving one linear
equation and one equation of order two The strategy is pretty much the same as for two linear equations word problems. Always let x and y be the answers to the questions asked and be absolutely clear what they represent.
EXAMPLE 7
The sum of two numbers is 3 and the sum of their squares is 65. What are the numbers?
Solution Let x = one number Let y = the other number
x + y = 3 … (1) x 2 + y 2 = 65 … (2)
x + y = 3 … (1) y = (3 − x)
Into (2): x 2 + (3 − x ) 2 = 65 x 2 + 9 − 6x + x 2 = 65 2 x 2 − 6x − 56 = 0 x 2 − 3x − 28 = 0
(x + 4)(x − 7) = 0 x = −4, 7
x = – 4: y = (3 – x) = (3 – (– 4)) = 7 x = 7: y = (3 – x) = (3 – (7)) = – 4
The numbers are –4 and 7. EXAMPLE 8
A television screen has diagonals of length 100 cm. If the perimeter of the screen is 280 cm, fi nd the length, breadth and area of the screen.
Solution Let x = length of screen Let y = breadth of screen
Perimeter: 2x + 2y = 280 x + y = 140 … (1)
Diagonals: x 2 + y 2 = 10 000 … (2) x + y = 140 … (1) y = (140 − x)
Into (2): A x 2 + (140 − x ) 2 = 10 000
x 2 + 19 600 − 280x + x 2 = 10 000 2 x 2 − 280x + 9600 = 0 x 2 − 140x + 4800 = 0 (x − 80)(x − 60) = 0 x = 80, 60
x = 80: y = (140 – x) = (140 – 80) = 60 x = 60: y = (140 – x) = (140 – 60) = 80
Solutions: length = 80 cm, breadth = 60 cm or length = 60 cm, breadth = 80 cm Area = 80 × 60 = 4800 c m 2
EXERCISE 7
(A) CONCEPTS AND SKILLS Solve the following for x and y:
