(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 rand...
Question 1 Report
(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.
Answer Details
(a) \(3y-2x=21\) and \(4y+5x=5\). Multiply the first by \(5\) and the second by \(2\): \[15y-10x=105,\qquad 8y+10x=10.\] Adding: \(23y=115\Rightarrow y=5\). Then \(3(5)-2x=21\Rightarrow-2x=6\Rightarrow x=-3\). So \(x=-3,\;y=5\).
(b) Card \(6\) wins \(\$60\); cards \(1\text{–}5\) lose \(\$10\times(\text{number})\).
- (i) A loss occurs whenever cards \(1,2,3,4,5\) are picked: \[P(\text{losing})=\frac{5}{6}.\]
- (ii) Losing exactly \(\$20\) in two picks: the outcomes per card are \(-10,-20,-30,-40,-50,+60\). The only pair of picks totalling \(-\$20\) is card \(1\) followed by card \(1\) \((-10-10=-20)\). \[P=\frac16\times\frac16=\frac{1}{36}.\]
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