Difference between revisions of "2018 AMC 10A Problems/Problem 12"

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== Problem ==
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How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations?
 
How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations?
<math>x+3y=3 \ \big||x|-|y|\big|=1</math>
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<cmath>x+3y=3</cmath>
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<cmath>\big||x|-|y|\big|=1</cmath>
 
<math>\textbf{(A) } 1 \qquad  
 
<math>\textbf{(A) } 1 \qquad  
 
\textbf{(B) } 2 \qquad  
 
\textbf{(B) } 2 \qquad  
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\textbf{(D) } 4 \qquad  
 
\textbf{(D) } 4 \qquad  
 
\textbf{(E) } 8 </math>
 
\textbf{(E) } 8 </math>
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== Solution ==
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The graph looks something like this:
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<asy>
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draw((-3,0)--(3,0), Arrows);
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draw((0,-3)--(0,3), Arrows);
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draw((2,3)--(0,1)--(-2,3), blue);
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draw((-3,2)--(-1,0)--(-3,-2), blue);
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draw((-2,-3)--(0,-1)--(2,-3), blue);
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draw((3,-2)--(1,0)--(3,2), blue);
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draw((-3,2)--(3,0), red);
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dot((-3,2));
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dot((3,0));
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dot((0,1));
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</asy>
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Now it's clear that there are <math>\boxed{3}</math> intersection points. (pinetree1)
  
 
== See Also ==
 
== See Also ==
  
 
{{AMC10 box|year=2018|ab=A|num-b=11|num-a=13}}
 
{{AMC10 box|year=2018|ab=A|num-b=11|num-a=13}}
 +
{{AMC12 box|year=2018|ab=A|num-b=9|num-a=11}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:30, 8 February 2018

Problem

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[x+3y=3\] \[\big||x|-|y|\big|=1\] $\textbf{(A) } 1 \qquad  \textbf{(B) } 2 \qquad  \textbf{(C) } 3 \qquad  \textbf{(D) } 4 \qquad  \textbf{(E) } 8$

Solution

The graph looks something like this: [asy] draw((-3,0)--(3,0), Arrows); draw((0,-3)--(0,3), Arrows); draw((2,3)--(0,1)--(-2,3), blue); draw((-3,2)--(-1,0)--(-3,-2), blue); draw((-2,-3)--(0,-1)--(2,-3), blue); draw((3,-2)--(1,0)--(3,2), blue); draw((-3,2)--(3,0), red); dot((-3,2)); dot((3,0)); dot((0,1)); [/asy]

Now it's clear that there are $\boxed{3}$ intersection points. (pinetree1)

See Also

2018 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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
2018 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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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