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

(→‎Solution 2)
(→‎Solution 2)
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<math>|3y-3-y| = |2y-3| = 1</math>
 
<math>|3y-3-y| = |2y-3| = 1</math>
−
     Subcase 1: <math>y>3/2</math>
+
     Subcase 1: <math>y>\frac{3}{2}</math>
 
<math>2y-3</math> is positive so <math>2y-3 = 1</math> and <math>y = 2</math> and <math>x = 3-3(2) = -3</math>
 
<math>2y-3</math> is positive so <math>2y-3 = 1</math> and <math>y = 2</math> and <math>x = 3-3(2) = -3</math>
−
     Subcase 2: <math>1<y<3/2</math>
+
     Subcase 2: <math>1<y<\frac{3}{2}</math>
 
<math>2y-3</math> is negative so <math>|2y-3| = 3-2y = 1</math>. <math>2y = 2</math> and so there are no solutions (<math>y</math> can't equal to <math>1</math>)
 
<math>2y-3</math> is negative so <math>|2y-3| = 3-2y = 1</math>. <math>2y = 2</math> and so there are no solutions (<math>y</math> can't equal to <math>1</math>)
  
Line 53: Line 53:
  
 
<math>3-3y</math> will be positive so <math>|3-3y-y| = |3-4y| = 1</math>
 
<math>3-3y</math> will be positive so <math>|3-3y-y| = |3-4y| = 1</math>
−
     Subcase 1: <math>y>4/3</math>
+
     Subcase 1: <math>y>\frac{4}{3}</math>
 
<math>3-4y</math> will be negative so <math>4y-3 = 1</math> \rightarrow <math>4y = 4</math>. There are no solutions (again, <math>y</math> can't equal to <math>1</math>)
 
<math>3-4y</math> will be negative so <math>4y-3 = 1</math> \rightarrow <math>4y = 4</math>. There are no solutions (again, <math>y</math> can't equal to <math>1</math>)
−
     Subcase 2: y<4/3
+
     Subcase 2: <math>y<\frac{4}{3}</math>
 
<math>3-4y</math> will be positive so <math>3-4y = 1</math> \rightarrow <math>4y = 2</math>. <math>y = \frac{1}{2}</math> and <math>x = \frac{3}{2}</math>.
 
<math>3-4y</math> will be positive so <math>3-4y = 1</math> \rightarrow <math>4y = 2</math>. <math>y = \frac{1}{2}</math> and <math>x = \frac{3}{2}</math>.
  

Revision as of 22:46, 8 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

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's 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.

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

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