Question

Harry had $32. He spent all the money on buying 3 notebooks for $x each and 4 packs of index cards for $y each. If Harry had bought 5 notebooks and 5 packs of index cards, he would have run short of $18.

A student concluded that the price of each notebook is $5 and the price of each pack of index cards is $1. Which statement best justifies whether the student's conclusion is correct or incorrect?

a. The student's conclusion is incorrect because the solution to the system of equations 3x + 4y = 32 and 5x + 5y = 50 is (8, 2).
b. The student's conclusion is incorrect because the solution to the system of equations 3x + 4y = 32 and 5x + 5y = 18 is (8, 2).
c. The student's conclusion is correct because the solution to the system of equations 3x + 4y = 32 and 5x + 5y = 18 is (5, 1).
d. The student's conclusion is correct because the solution to the system of equations 3x − 4y = 32 and 5x − 5y = 50 is (5, 1).

Answer 1

a. The student's conclusion is incorrect because the solution to the system of equations 3x + 4y = 32 and 5x + 5y = 50 is (8, 2).

Explanation:

The second equation needs to be ...

5x +5y = 50

because the wording "short of $18" means that the actual cost of that purchase would be $18 more than the $32 that Harry had. 18+32 = 50.

The only answer choice that shows this as the second equation is choice A.

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In fact, the solution to the system of equations is (8, 2), as can be seen in the graph below.