2002 AMC 10B Problem 13 — Solution | LIVE by Po-Shen Loh

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2002 AMC 10B Problem 13

Attempt Problem 13 of the 2002 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2002 AMC 10B solutions, or check the answer key.

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13.

What value of $x$ makes $8xy - 12y + 2x - 3 = 0$ true for all values of $y?$

$\dfrac23$

$\dfrac32$ or $-\dfrac14$

$-\dfrac23$ or $-\dfrac14$

$\dfrac32$

$-\dfrac32$ or $-\dfrac14$

Answer: D

Concepts:factoring algebraic manipulation

Difficulty rating: 1220

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.

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