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

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

Solutions

Solution 1

We can solve this by graphing the equations. The second equation looks challenging to graph, but start by graphing it in the first quadrant only (which is easy since the inner absolute value signs can be ignored), then simply reflect that graph into the other quadrants.

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 $\boxed{\textbf{(C) } 3}$ 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 $\boxed{\textbf{(C) } 3}$ . Danny Li JHS $\text{\LaTeX}$ edit by pretzel, very minor $\text{\LaTeX}$ edits by Bryanli

Solution 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: $(0, 1)$ , $(-3,2)$ , $(1.5, 0.5)$ so the answer is $\boxed{\textbf{(C) } 3}$

~trumpeter

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.

@@ Line 55: / Line 55: @@
 we obtain the three solutions: <math>(0, 1)</math>, <math>(-3,2)</math>, <math>(1.5, 0.5)</math> so the answer is <math>\boxed{\textbf{(C) } 3}</math>
-~ccx09
+~trumpeter
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