Difference between revisions of "2018 AMC 10A Problems/Problem 12"
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+ | == 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 18:30, 8 February 2018
Problem
How many ordered pairs of real numbers satisfy the following system of equations?
Solution
The graph looks something like this:
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.