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B. 15/12

C. 180

D. 12/15

First find the value of

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A. 100(0.085)**[**1.085*x* + *y*
+ 1.080*z***]** ≥ 1000

B. 100(1.085)**[**0.85*x* + *y*
+ 0.80*z***]** ≤ 1000

C. 100(1.085)**[**0.85*x* + *y*
+ 0.80*z***]** ≥ 1000

D. 100(0.085)**[**0.85*x* + *y*
+ 0.80*z***] **≥ 1000

A senior citizen gets a 20% discount, so his
rate is 100(1 – 0.20) = 100(0.80) per night of stay. After tax, a senior
citizen will pay 100(0.80)(1 + 0.085) = 100(0.80)(1.085) dollars per night. Thus
the motel will receive 100(0.80)(1.085)*z*
from senior citizens if *z* senior
citizens stay per night. An adult gets no discount, so they will pay 100(1 +
0.085) = 100(1.085) dollars per night. A total of 100(1.085)*y* is collected per night by the motel if
*y* adults stay per night. A child
receives a 15% discount, so his rate including tax is 100(1 – 0.15)(1 + 0.085)
= 100(0.85)(1.085) dollars per night. The motel will collect 100(0.85)(1.085)*x* if *x*
children stay per night. Thus the total collected by the motel for *x* children, *y* adults, and *z* senior
citizens staying for the night is 100(0.85)(1.085)*x* + 100(1.085)*y* + 100(0.80)(1.085)*z* = 100(1.085)**[**0.85*x* + *y* + 0.80*z***] **dollars.
Since this sum must be at least 1000 dollars, it must be greater than or equal
to 1000. Thus the correct answer is (C).

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A. 6

B. -12

C. 1/3

D. -1/12

First try to think graphically. Each of the equations above is linear. A solution to a system of two linear equations means that there is a point of intersection between the two lines corresponding to the two equations (there can only be one such point). Having no such point means that the lines are parallel (they never meet at any point). If the lines are parallel, they must have the same slope. To have the same slope, coefficients in front of *z* in each equation are such that 2 = 5*k*/8 for some constant *k*. In other words, the constant *k* transforms a variable coefficient in one equation into the same exact variable coefficient in the second equation. This means that 2 is a scalar multiple of 5/8 for some *k*. Solving for *k* we get *k* = 16/5. Similarly, coefficients in front of *w* are such that -4/3 = 5*A* • *k*, and using the known *k* we have -4/3 = 5*A* • 16/5, which gives *A* = -1/12. Thus the correct answer is (D).

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