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

(Solution 2)
(Solution 3 (do not use in the real test))
Line 48: Line 48:
 
List all of the cases out.
 
List all of the cases out.
  
<math>x+3y=3</math>
+
<math>(0, 1)</math>
  
$||x|-|y||=1
+
<math>(-2,3)</math>
 +
 
 +
<math>(1.5, 0.5)</math>
 +
 
 +
We can see that there is 3 solutions, so the answer is <math>C</math>
 +
 
 +
-Baolan
  
 
==See Also==
 
==See Also==

Revision as of 15:25, 9 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 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: $y>\frac{3}{2}$

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

   Subcase 2: $1<y<\frac{3}{2}$

$2y-3$ 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: $y>\frac{4}{3}$

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

   Subcase 2: $y<\frac{4}{3}$

$3-4y$ 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.

$(0, 1)$

$(-2,3)$

$(1.5, 0.5)$

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

-Baolan

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS