Difference between revisions of "2006 AMC 12B Problems/Problem 8"

 
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== Problem ==
 
== Problem ==
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<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>The lines <math>x = \frac 14y + a</math> and <math>y = \frac 14x + b</math> intersect at the point <math>(1,2)</math>.  What is <math>a + b</math>?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude>
  
== Solution ==
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<math>
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\text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94
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</math>
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== Solution 1 ==
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<math>4x-4a=y</math>
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<math>4x-4a=\frac{1}{4}x+b</math>
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<math>4\cdot1-4a=\frac{1}{4}\cdot1+b=2</math>
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<math>a=\frac{1}{2}</math>
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<math>b=\frac{7}{4}</math>
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<math>a+b=\frac{9}{4} \Rightarrow \text{(E)}</math>
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== Solution 2 ==
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Add both equations:
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<math>\begin{cases}x=\frac{1}{4}y+a\\ y=\frac{1}{4}x+b \end{cases}</math>
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Simplify:
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<math>\frac{4}{4}(x+y)=\frac{1}{4}(x+y)+(a+b)</math>
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Isolate our solution:
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<math>\frac{3}{4}(x+y)=a+b</math>
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Substitute the point of intersection <math>[x=1, y=2]</math>
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<math>a+b=\frac{3}{4}\cdot3=\frac{9}{4} \Rightarrow \text{(E)}</math>
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==Solution 3==
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Plugging in <math>(1,2)</math> into the first equation, and solving for <math>a</math> we get <math>a</math> as <math>\frac{1}{2}</math>.
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Doing the same for the second equation for the second equation, we get <math>b</math> as <math>\frac{7}{4}</math>
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Adding <math>a+b = \frac{1}{2} + \frac{7}{4} = \frac{9}{4} \Rightarrow \text{(E)}</math>
  
 
== See also ==
 
== See also ==
* [[2006 AMC 12B Problems]]
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{{AMC12 box|year=2006|ab=B|num-b=7|num-a=9}}
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{{MAA Notice}}

Revision as of 15:16, 12 September 2019

Problem

The lines $x = \frac 14y + a$ and $y = \frac 14x + b$ intersect at the point $(1,2)$. What is $a + b$?

$\text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) } 2 \qquad \text {(E) } \frac 94$

Solution 1

$4x-4a=y$

$4x-4a=\frac{1}{4}x+b$

$4\cdot1-4a=\frac{1}{4}\cdot1+b=2$

$a=\frac{1}{2}$

$b=\frac{7}{4}$

$a+b=\frac{9}{4} \Rightarrow \text{(E)}$

Solution 2

Add both equations:

$\begin{cases}x=\frac{1}{4}y+a\\ y=\frac{1}{4}x+b \end{cases}$

Simplify:

$\frac{4}{4}(x+y)=\frac{1}{4}(x+y)+(a+b)$

Isolate our solution:

$\frac{3}{4}(x+y)=a+b$

Substitute the point of intersection $[x=1, y=2]$

$a+b=\frac{3}{4}\cdot3=\frac{9}{4} \Rightarrow \text{(E)}$

Solution 3

Plugging in $(1,2)$ into the first equation, and solving for $a$ we get $a$ as $\frac{1}{2}$.

Doing the same for the second equation for the second equation, we get $b$ as $\frac{7}{4}$

Adding $a+b = \frac{1}{2} + \frac{7}{4} = \frac{9}{4} \Rightarrow \text{(E)}$

See also

2006 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
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 12 Problems and Solutions

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