Difference between revisions of "2018 AMC 10A Problems/Problem 12"
| Line 1: | Line 1: | ||
| + | == Problem == | ||
| + | |||
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? | ||
| − | < | + | <cmath>x+3y=3</cmath> |
| + | <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 | ||
| Line 6: | Line 9: | ||
\textbf{(D) } 4 \qquad | \textbf{(D) } 4 \qquad | ||
\textbf{(E) } 8 </math> | \textbf{(E) } 8 </math> | ||
| + | |||
| + | == 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 <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 17:30, 8 February 2018
Problem
How many ordered pairs of real numbers 
satisfy the following system of equations?
![\[x+3y=3\]](http://latex.artofproblemsolving.com/2/b/6/2b68f21d4456251f723b5b9780794891f520a3c0.png)
![\[\big||x|-|y|\big|=1\]](http://latex.artofproblemsolving.com/d/9/1/d91f752f8219148671d39d50f1e9d6a726c3ce35.png)
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]](http://latex.artofproblemsolving.com/0/f/9/0f9bc2d077559bb70dba8f9ded81e499a12acb2f.png)
Now it's clear that there are 
intersection points. (pinetree1)
See Also
| 2018 AMC 10A (Problems • Answer Key • Resources) | ||
| 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 (Problems • Answer Key • Resources) | |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
