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Difference between revisions of "2002 AMC 10B Problems/Problem 13"

(2002 AMC 10B Problem 13)
 
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== Problem ==
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Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>.
 
Find the value(s) of <math>x</math> such that <math>8xy - 12y + 2x - 3 = 0</math> is true for all values of <math>y</math>.
  
 
<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac34 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math>
 
<math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac34 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14</math>
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== Solution ==
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We have <math>8xy - 12y + 2x - 3 = 4y(2x - 3) + (2x - 3) = (4y + 1)(2x - 3)</math>.
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As <math>(4y + 1)(2x - 3) = 0</math> must be true for all <math>y</math>, we must have <math>2x - 3 = 0</math>, hence <math>\boxed{x = \frac 32}</math>.
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(Too bad there is no such option -- maybe a typo when transcribing the options? Or the question is not formulated correctly? Note that the other fraction in option <math>B</math> would be the answer to the complementary question "find the value <math>y</math> such that ... for all <math>x</math>".)
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== See Also ==
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{{AMC10 box|year=2002|ab=B|num-b=12|num-a=14}}

Revision as of 06:35, 2 February 2009

Problem

Find the value(s) of $x$ such that $8xy - 12y + 2x - 3 = 0$ is true for all values of $y$.

$\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf{(C) } -\frac23 \text{ or } -\frac14 \qquad \textbf{(D) } \frac34 \qquad \textbf{(E) } -\frac32 \text{ or } -\frac14$

Solution

We have $8xy - 12y + 2x - 3 = 4y(2x - 3) + (2x - 3) = (4y + 1)(2x - 3)$.

As $(4y + 1)(2x - 3) = 0$ must be true for all $y$, we must have $2x - 3 = 0$, hence $\boxed{x = \frac 32}$.

(Too bad there is no such option -- maybe a typo when transcribing the options? Or the question is not formulated correctly? Note that the other fraction in option $B$ would be the answer to the complementary question "find the value $y$ such that ... for all $x$".)

See Also

2002 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
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