(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. = \(\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}\)