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

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:

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

   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 $\boxed{\textbf{(C) } 3}$ .

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

@@ Line 53: / Line 53: @@
 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 (Casework) ==
+== Solution 3 (Straightforward Casework) ==
-Note that <math>||x| - |y||</math> can take on one of four values: <math>x + y</math>, <math>x - y</math>, <math>-x + y</math>, <math>-x -y</math>. So we have 4 cases:
+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:
 '''Case 1: ||x| - |y|| = x+y  '''
@@ Line 118: / Line 118: @@
 ~BakedPotato66
+~<B+
 == Solution 4 ==
@@ Line 132: / Line 134: @@
 ~OlutosinNGA
-== Solution 6 (Desperate) ==
+== Video Solution 1 ==
-Since the absolute value is the square root of the square, we get that the first equation is quartic(degree <math>4</math>) and the other is linear. Subtract to get <math>\boxed{\textbf{(C) } 3}</math>.
-~integralarefun
-This doesn't work ~ hurdler
-For a more clear explanation of why it doesn't work:
-It doesn't work because the linear function doesn't need to pass through 3 points, it could pass through 0–4 points depending on their slopes, intercepts, etc. For example, a quadratic formula could have 0, 1, or 2 solutions, meaning it could have 0, 1, or 2 points of intersection. This is why the solution would not work. Good try though.
-~ Continuous_Pi
-== Video Solution ==
 https://youtu.be/o63EtwelFp0
 ~savannahsolver
+== Video Solution 2==
+https://www.youtube.com/watch?v=llMgyOkjNgU&ab_channel=TheBeautyofMath
 ==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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