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

 
(19 intermediate revisions by 13 users not shown)
Line 1: Line 1:
{{duplicate|[[2018 AMC 12A Problems|2018 AMC 12A #10]] and [[2018 AMC 10A Problems|2018 AMC 10A #12]]}}
+
{{duplicate|[[2018 AMC 10A Problems/Problem 12|2018 AMC 10A #12]] and [[2018 AMC 12A Problems/Problem 10|2018 AMC 12A #10]]}}
  
 
== Problem ==
 
== 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>\begin{align*}
<cmath>\big||x|-|y|\big|=1</cmath>
+
x+3y&=3 \
 +
\big||x|-|y|\big|&=1
 +
\end{align*}</cmath>
 
<math>\textbf{(A) } 1 \qquad  
 
<math>\textbf{(A) } 1 \qquad  
 
\textbf{(B) } 2 \qquad  
 
\textbf{(B) } 2 \qquad  
Line 12: Line 13:
 
\textbf{(E) } 8 </math>
 
\textbf{(E) } 8 </math>
  
==Solutions==
+
== Solution 1 ==
===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.
 
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.
  
Line 31: Line 31:
 
Now, it becomes clear that there are <math>\boxed{\textbf{(C) } 3}</math> intersection points.
 
Now, it becomes clear that there are <math>\boxed{\textbf{(C) } 3}</math> intersection points.
  
===Solution 2===
+
== Solution 2 ==
 
<math>x+3y=3</math> can be rewritten to <math>x=3-3y</math>. Substituting <math>3-3y</math> for <math>x</math> in the second equation will give <math>||3-3y|-y|=1</math>. Splitting this question into casework for the ranges of <math>y</math> will give us the total number of solutions.  
 
<math>x+3y=3</math> can be rewritten to <math>x=3-3y</math>. Substituting <math>3-3y</math> for <math>x</math> in the second equation will give <math>||3-3y|-y|=1</math>. Splitting this question into casework for the ranges of <math>y</math> will give us the total number of solutions.  
  
Line 47: Line 47:
 
<math>\textbf{Case 3:}</math> <math>y<1</math>:
 
<math>\textbf{Case 3:}</math> <math>y<1</math>:
 
<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>\frac{4}{3}</math>
+
     Subcase 1: <math>y>\frac{3}{4}</math>
<math>3-4y</math> will be negative so <math>4y-3 = 1</math> <math>\rightarrow</math> <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> <math>\rightarrow</math> <math>4y = 4</math>. We already have this solution from Case 2 as <math>y = 1</math>.
     Subcase 2: <math>y<\frac{4}{3}</math>
+
     Subcase 2: <math>y<\frac{3}{4}</math>
 
<math>3-4y</math> will be positive so <math>3-4y = 1</math> <math>\rightarrow</math> <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> <math>\rightarrow</math> <math>4y = 2</math>. <math>y = \frac{1}{2}</math> and <math>x = \frac{3}{2}</math>.
 
Thus, the solutions are: <math>(-3,2), (0,1), \left(\frac{3}{2},\frac{1}{2} \right)</math>, and the answer is <math>\boxed{\textbf{(C) } 3}</math>.
 
Thus, the solutions are: <math>(-3,2), (0,1), \left(\frac{3}{2},\frac{1}{2} \right)</math>, and the answer is <math>\boxed{\textbf{(C) } 3}</math>.
  
===Solution 3===
+
== Solution 3 (Straightforward Casework) ==
Note that <math>||x| - |y||</math> can take on either of four values: <math>x + y</math>, <math>x - y</math>, <math>-x + y</math>, <math>-x -y</math>.
+
Note that <math>||x| - |y||</math> can take on one of four values where <math>x</math> and <math>y</math> are: <math>x + y</math> (positive, positive), <math>x - y</math> (positive, negative), <math>-x + y</math> (negative, positive), <math>-x -y</math> (negative, negative). So we have 4 cases which we solve by cancellation with the two equations from the problem:
Solving the equations (by elimination, either adding the two equations or subtracting),
 
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>. One of those equations overlap into <math>(0, 1)</math> so there's only 3 solutions.
 
  
~trumpeter, ccx09
+
'''Case 1: ||x| - |y|| = x+y  '''
~minor edit, XxHalo711
+
<cmath>x+3y=3</cmath>
 +
<cmath>x+y=1</cmath>
  
===Solution 4===
+
Subtracting:
 +
 
 +
<math>2y=2 \Rightarrow y=1</math> and <math>x=0</math>.
 +
 
 +
<math>\text{Result: } (0,1)</math>
 +
 
 +
 
 +
'''Case 2: ||x| - |y|| = x-y'''
 +
<cmath>x+3y=3</cmath>
 +
<cmath>x-y=1</cmath>
 +
 
 +
Subtacting:
 +
 
 +
<math>4y=2 \Rightarrow y=\dfrac{1}{2}</math> an <math>x=\dfrac{3}{2}</math>.
 +
 
 +
<math>\text{Result: } \left(\dfrac{3}{2},\dfrac{1}{2}\right)</math>
 +
 
 +
 
 +
'''Case 3: ||x| - |y|| = -x+y'''
 +
<cmath>x+3y=3</cmath>
 +
<cmath>-x+y=1</cmath>
 +
 
 +
Adding:
 +
 
 +
<math>4y=4 \Rightarrow y=1</math> and <math>x=0</math>.
 +
 
 +
 
 +
<math>\text{Result: } (0,1)</math>. Since this is the same solution as we got in Case 1, we can not count this (otherwise, we would have overcounted).
 +
 
 +
 
 +
'''Case 4: ||x| - |y|| = -x-y'''
 +
<cmath>x+3y=3</cmath>
 +
<cmath>-x-y=1</cmath>
 +
 
 +
Adding:
 +
 
 +
<math>2y=4 \Rightarrow y=2</math> and <math>x=-3</math>.
 +
 
 +
 
 +
<math>\text{Result: } (-3,2)</math>.
 +
 
 +
 
 +
'''Answer'''
 +
 
 +
We solved each case by elimination (either adding the two equations or subtracting), to obtain three solutions: <math>(0, 1)</math>, <math>(-3,2)</math>, <math>\left(\dfrac{3}{2}, \dfrac{1}{2}\right)</math>.
 +
 
 +
Our answer is <math>\boxed{\textbf{(C) } 3}</math>.
 +
 
 +
 
 +
'''Contributors'''
 +
 
 +
~trumpeter
 +
 
 +
~ccx09
 +
 
 +
~mXxHalo711
 +
 
 +
~BakedPotato66
 +
 
 +
~<B+
 +
 
 +
== Solution 4 ==
 
Just as in solution <math>2</math>, we derive the equation <math>||3-3y|-|y||=1</math>. If we remove the absolute values, the equation collapses into four different possible values. <math>3-2y</math>, <math>3-4y</math>, <math>2y-3</math>, and <math>4y-3</math>, each equal to either <math>1</math> or <math>-1</math>. Remember that if <math>P-Q=a</math>, then <math>Q-P=-a</math>. Because we have already taken <math>1</math> and <math>-1</math> into account, we can eliminate one of the conjugates of each pair, namely <math>3-2y</math> and <math>2y-3</math>, and <math>3-4y</math> and <math>4y-3</math>. Find the values of <math>y</math> when <math>3-2y=1</math>, <math>3-2y=-1</math>, <math>3-4y=1</math> and <math>3-4y=-1</math>. We see that <math>3-2y=1</math> and <math>3-4y=-1</math> give us the same value for <math>y</math>, so the answer is <math>\boxed{\textbf{(C) } 3}</math>
 
Just as in solution <math>2</math>, we derive the equation <math>||3-3y|-|y||=1</math>. If we remove the absolute values, the equation collapses into four different possible values. <math>3-2y</math>, <math>3-4y</math>, <math>2y-3</math>, and <math>4y-3</math>, each equal to either <math>1</math> or <math>-1</math>. Remember that if <math>P-Q=a</math>, then <math>Q-P=-a</math>. Because we have already taken <math>1</math> and <math>-1</math> into account, we can eliminate one of the conjugates of each pair, namely <math>3-2y</math> and <math>2y-3</math>, and <math>3-4y</math> and <math>4y-3</math>. Find the values of <math>y</math> when <math>3-2y=1</math>, <math>3-2y=-1</math>, <math>3-4y=1</math> and <math>3-4y=-1</math>. We see that <math>3-2y=1</math> and <math>3-4y=-1</math> give us the same value for <math>y</math>, so the answer is <math>\boxed{\textbf{(C) } 3}</math>
  
 
~Zeric Hang
 
~Zeric Hang
  
===Solution 5===
+
== Solution 5 ==
 
Just as in solution <math>2</math>, we derive the equation <math>x=3-3y</math>. Squaring both sides in the second equation gives <math>x^2+y^2-2|xy|=1</math>. Putting <math>x=3-3y</math> and doing a little calculation gives <math>10y^2-18y+9-2|3y-3y^2|=1</math>. From here we know that <math>3y-3y^2</math> is either positive or negative.
 
Just as in solution <math>2</math>, we derive the equation <math>x=3-3y</math>. Squaring both sides in the second equation gives <math>x^2+y^2-2|xy|=1</math>. Putting <math>x=3-3y</math> and doing a little calculation gives <math>10y^2-18y+9-2|3y-3y^2|=1</math>. From here we know that <math>3y-3y^2</math> is either positive or negative.
  
Line 74: Line 134:
 
~OlutosinNGA
 
~OlutosinNGA
  
==Video Solution==
+
== Video Solution 1 ==
 
https://youtu.be/o63EtwelFp0
 
https://youtu.be/o63EtwelFp0
  
 
~savannahsolver
 
~savannahsolver
 +
 +
== Video Solution 2==
 +
https://www.youtube.com/watch?v=llMgyOkjNgU&ab_channel=TheBeautyofMath
  
 
==See Also==
 
==See Also==
Line 84: Line 147:
 
{{MAA Notice}}
 
{{MAA Notice}}
  
[[Category:Intermediate Algebra Problems]]
+
[[Category:Introductory Algebra Problems]]

Latest revision as of 08:45, 14 October 2024

The following problem is from both the 2018 AMC 10A #12 and 2018 AMC 12A #10, so both problems redirect to this page.

Problem

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

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.

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

$3-4y$ will be negative so $4y-3 = 1$ $\rightarrow$ $4y = 4$. We already have this solution from Case 2 as $y = 1$.

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

$3-4y$ 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 $\boxed{\textbf{(C) } 3}$.

Solution 3 (Straightforward Casework)

Note that $||x| - |y||$ can take on one of four values where $x$ and $y$ are: $x + y$ (positive, positive), $x - y$ (positive, negative), $-x + y$ (negative, positive), $-x -y$ (negative, negative). So we have 4 cases which we solve by cancellation with the two equations from the problem:

Case 1: ||x| - |y|| = x+y \[x+3y=3\] \[x+y=1\]

Subtracting:

$2y=2 \Rightarrow y=1$ and $x=0$.

$\text{Result: } (0,1)$


Case 2: ||x| - |y|| = x-y \[x+3y=3\] \[x-y=1\]

Subtacting:

$4y=2 \Rightarrow y=\dfrac{1}{2}$ an $x=\dfrac{3}{2}$.

$\text{Result: } \left(\dfrac{3}{2},\dfrac{1}{2}\right)$


Case 3: ||x| - |y|| = -x+y \[x+3y=3\] \[-x+y=1\]

Adding:

$4y=4 \Rightarrow y=1$ and $x=0$.


$\text{Result: } (0,1)$. Since this is the same solution as we got in Case 1, we can not count this (otherwise, we would have overcounted).


Case 4: ||x| - |y|| = -x-y \[x+3y=3\] \[-x-y=1\]

Adding:

$2y=4 \Rightarrow y=2$ and $x=-3$.


$\text{Result: } (-3,2)$.


Answer

We solved each case by elimination (either adding the two equations or subtracting), to obtain three solutions: $(0, 1)$, $(-3,2)$, $\left(\dfrac{3}{2}, \dfrac{1}{2}\right)$.

Our answer is $\boxed{\textbf{(C) } 3}$.


Contributors

~trumpeter

~ccx09

~mXxHalo711

~BakedPotato66

~<B+

Solution 4

Just as in solution $2$, we derive the equation $||3-3y|-|y||=1$. If we remove the absolute values, the equation collapses into four different possible values. $3-2y$, $3-4y$, $2y-3$, and $4y-3$, each equal to either $1$ or $-1$. Remember that if $P-Q=a$, then $Q-P=-a$. Because we have already taken $1$ and $-1$ into account, we can eliminate one of the conjugates of each pair, namely $3-2y$ and $2y-3$, and $3-4y$ and $4y-3$. Find the values of $y$ when $3-2y=1$, $3-2y=-1$, $3-4y=1$ and $3-4y=-1$. We see that $3-2y=1$ and $3-4y=-1$ give us the same value for $y$, so the answer is $\boxed{\textbf{(C) } 3}$

~Zeric Hang

Solution 5

Just as in solution $2$, we derive the equation $x=3-3y$. Squaring both sides in the second equation gives $x^2+y^2-2|xy|=1$. Putting $x=3-3y$ and doing a little calculation gives $10y^2-18y+9-2|3y-3y^2|=1$. From here we know that $3y-3y^2$ is either positive or negative.

When positive, we get $2y^2-3y+1=0$ and then, $y=1/2$ or $y=1$. When negative, we get $y^2-3y+2=0$ and then, $y=2$ or $y=1$. Clearly, there are $3$ different pairs of values and that gives us $\boxed{\textbf{(C) } 3}$

~OlutosinNGA

Video Solution 1

https://youtu.be/o63EtwelFp0

~savannahsolver

Video Solution 2

https://www.youtube.com/watch?v=llMgyOkjNgU&ab_channel=TheBeautyofMath

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