2002 AMC 10B Problem 13

Question

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

Group terms to factor the left side

8xy - 12y + 2x - 3

= (4y + 1)(2x - 3)

; it must vanish for every

y

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.

Problem 13 in other years