(a) Solve the simultaneous equations 3y – 2x = 21 ; 4y + 5x = 5.
(b) Six identical cards numbered 1 – 6 are placed face down. A card is to be picked at random. A person wins $60.00 if he picks the card numbered 6. If he picks any of the other cards, he loses $10.00 times the number on the card. Calculate the probability of (i) losing ; (ii) losing $20.00 after two picks.
Explanation
(a) (3y – 2x = 21 … (1))
(4y + 5x = 5 …. (2))
Multiply (1) by 4 and (2) by 3 so we have,
(12y – 8x = 84 … (3))
(12y + 15x = 15 … (4))
(3) – (4) : (-8x – 15x = 69 implies -23x = 69)
(x = frac{69}{-23} = -3)
Put x = -3 in (1),
(3y – 2(-3) = 21)
(3y + 6 = 21 implies 3y = 21 – 6 = 15)
(y = frac{15}{3} = 5)
(x, y = -3, 5).
(b)(i) Probability of losing
Probability of winning = (frac{1}{6})
(therefore) Probability of losing = (1 – frac{1}{6} = frac{5}{6}).
(ii) Losing $20.00 after two picks = picking the card numbered 1 twice.
= (frac{1}{6} times frac{1}{6} = frac{1}{36})