2002 AMC 10B Problem 13

Problem 13 of 25IntermediateAlgebra

What value of

x

makes

8xy - 12y + 2x - 3 = 0

true for all values of

y?

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Solution

Grouping and factoring,

\begin{aligned} &8xy - 12y + 2x - 3 \\ &= 4y(2x - 3) + (2x - 3) \\ &= (4y + 1)(2x - 3). \end{aligned}

For this to equal

0

for all

y,

the factor that depends on

y

cannot be forced to zero, so we need

2x - 3 = 0,

giving

x = \dfrac32.

Thus, the correct answer is D.

Concepts: factoring · algebraic manipulation

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