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

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2002 AMC 10B Problem 13

Below is the professionally curated solution for Problem 13 of the 2002 AMC 10B, 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.

All of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:factoring algebraic manipulation

Difficulty rating: 1220

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 $-\dfrac14$

Solution:

Grouping and factoring, $8xy - 12y + 2x - 3 = 4y(2x - 3) + (2x - 3) = (4y + 1)(2x - 3).$

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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