MATHS...simultaneous equation

72.
(a)
[(2400/x) + 50](x - 1) = 3000
(2400/x)x - (2400/x) + 50x - 50 = 3000
2400 - (2400/x) + 50x - 50 = 3000
50x - 650 - (2400/x) = 0
50[x - 13 - (48/x)] = 0
x - 13 - (48/x) = 0
x[x - 13 - (48/x)] = 0
x² - 13x - 48 = 0

(b)
x² - 13x - 48 = 0
(x - 16)(x + 3) = 0
x = 16 or x = -3 (rejected)

Selling price of each calculator
= $[(2400/16) + 50]
= $200


73.
Side of square ABCD:
x + y = 8
y = 8 - x …… (1)

(Area of ΔAMD) + (Area of ΔCND) = Area of ΔBMN
(1/2)8x + (1/2)8x = (1/2)y²
8x + 8x = y²
16x = y² …… (2)

Put (1) into (2):
16x = (8 - x)²
16x = 64 - 16x + x²
x² - 32x + 64 = 0
x = {32 ± √[(-32)² - 4(1)(64)]}/2
x = 16 ± 8√3
x = 29.86 (rejected) or x = 2.14 (2 decimal places)


74.
(a)
PB = 10 - x

In ΔPBQ:
PQ² = PB² + BQ² (Pythagorean theorem)
y² = (10 - x)² + x²
y² = 100 - 20x + x² + x²
y² = 2x² - 20x + 100
y = √(2x² - 20x + 100)

(b)
Area of sq. PQRS = 10(Area of ΔPBQ)
y² = 10(1/2)x(10 - x)
y² = 50x - 5x² …… (1)

In (a):
y = √(2x² - 20x + 100)
y² = 2x² - 20x + 100 …… (2)

(2) = (1):
2x² - 20x + 100 = 50x - 5x²
7x² - 70x + 100 = 0
x = {70 ± √[(-70)² - 4(7)(100)]}/2(7)
x = 8.27 or x = 1.73
(2 decimal places)