2018 AMC 10A Problems/Problem 12

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 1

The graph looks something like this: $[asy] draw((-3,0)--(3,0), Arrows); draw((0,-3)--(0,3), Arrows); draw((2,3)--(0,1)--(-2,3), blue); draw((-3,2)--(-1,0)--(-3,-2), blue); draw((-2,-3)--(0,-1)--(2,-3), blue); draw((3,-2)--(1,0)--(3,2), blue); draw((-3,2)--(3,0), red); dot((-3,2)); dot((3/2,1/2)); dot((0,1)); [/asy]$ Now, it becomes clear that there are $\boxed{3}$ intersection points. (pinetree1)

Solution 2

$x+3y=3$ can be rewritten to $x=3-3y$ . Substituting $3-3y$ for $x$ in the second equation will give $||3-3y|-y|=1$ . Splitting this question into casework for the ranges of y will give us the total number of solutions.

$\textbf{Case 1:}$ $y>1$ $3-3y$ will be negative so $|3-3y| = 3y-3.$ $|3y-3-y| = |2y-3| = 1$

   Subcase 1:

$2y-3$ is positive so $2y-3 = 1$ and $y = 2$ and $x = 3-3(2) = -3$

   Subcase 2:

$2y-3$ is negative so $|2y-3| = 3-2y = 1$ . $2y = 2$ and so there are no solutions ( $y$ can't equal to $1$ )

$\textbf{Case 2:}$ $y = 1$ It is fairly clear that $x = 0.$

$\textbf{Case 3:}$ $y<1$ $3-3y$ will be positive so $|3-3y-y| = |3-4y| = 1$

   Subcase 1:

$3-4y$ will be negative so $4y-3 = 1$ \rightarrow $4y = 4$ . There are no solutions (again, $y$ can't equal to $1$ )

   Subcase 2:

$3-4y$ will be positive so $3-4y = 1$ \rightarrow $4y = 2$ . $y = \frac{1}{2}$ and $x = \frac{3}{2}$ . Thus, the solutions are: $(-3,2), (0,1), \left(\frac{3}{2},\frac{1}{2} \right)$ , and the answer is $3,$ or $\boxed{\textbf{(C)}}$ Solution by Danny Li JHS, $\text{\LaTeX}$ edit by pretzel.

Solution 3 (do not use in the real test)

List all of the cases out.

, ,

We can see that there is 3 solutions, so the answer is $C$

-Baolan

Solution 4 (Similar to 3)

Note that ||x| - |y|| can take on either of four values: x + y, x - y, -x + y, -x -y. Solving the equations (by elimination, either adding the two equations or subtracting), we obtain the three solutions: $(0, 1)$ , $(-2,3)$ , $(1.5, 0.5)$ so the answer is $C$ .

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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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.

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