Let y = 2g + x² and g = 5x + 1. If g² = 256, then which of the following can be a value of 2y + 3g? A. 100 B. 130 C. 141 D. 142 E. 160

(B) First note that g can either be 16 or -16. We are going to have to use both to find all solutions before we can choose a final answer. We begin by substituting g = 16 and then g = -16. If g = 16, then x = 3. If g = -16, then x = -17/5. Since all of our solutions are whole numbers, we can dismiss -17/5. We use g = 16 and x = 3 to determine the value of y: y = 2(16) + 9 = 41. Finally, we substitute the values for y and g into the last expression to determine its value : 2(41) + 3(16) = 82 + 48 = 130; the correct answer is (B). ---------- Hello, How would you know there could be no non- whole numbers as solutions in this problem? Please explain. Thank you.

How would you know there could be no non- whole numbers as solutions in this problem?

If we consider the question statement alone, then there can be two values of 2y + 3g. One value for g = 16 and one value for g = -16.

However, we have not only the question statement itself but five answer choices as well. The question statement asks ".. which of the following can be ...". So one (only one) of the two values must be among the answer choices.

All the answer choices are integers. As wee see in the explanation, g = 16 yields that value, 130. That is enough and we do NOT need to consider g = -16.

However, if we calculate x = -17/5 for g = -16 it is clear that y = 2g + x² = 2 × integer + fraction² = fraction so 2y + 3g = 2 × fraction + 3 × integer = fraction

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