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Difference between revisions of "2021 Fall AMC 10A Problems/Problem 14"

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How many ordered pairs <math>(x,y)</math> of real numbers satisfy the following system of equations?  
 
How many ordered pairs <math>(x,y)</math> of real numbers satisfy the following system of equations?  
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<cmath>x^2+3y=9</cmath>
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<cmath>\begin{align*}
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<cmath>(|x|+|y|-4)^2 = 1</cmath>
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x^2+3y&=9 \\
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(|x|+|y|-4)^2 &= 1
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\end{align*}</cmath>
 +
<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 7</math>
  
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<math>\textbf{(A )} 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 7</math>
+
==Solution 1 (Graphing)==
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−
 
 
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==Solution (Graphing)==
 
−
 
 
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The second equation is <math>(|x|+|y| - 4)^2 = 1</math>. We know that the graph of <math>|x| + |y|</math> is a very simple diamond shape, so let's see if we can reduce this equation to that form.
 
−
 
 
−
<math>(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5</math>}.
 
  
 +
The second equation is <math>(|x|+|y| - 4)^2 = 1</math>. We know that the graph of <math>|x| + |y|</math> is a very simple diamond shape, so let's see if we can reduce this equation to that form: <cmath>(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5\}.</cmath>
 
We now have two separate graphs for this equation and one graph for the first equation, so let's put it on the coordinate plane:
 
We now have two separate graphs for this equation and one graph for the first equation, so let's put it on the coordinate plane:
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−
 
 
<asy>
 
<asy>
  
Line 88: Line 83:
 
}  
 
}  
 
draw(graph(f,-5,5));
 
draw(graph(f,-5,5));
 +
</asy>
 +
We see from the graph that there are <math>5</math> intersections, so the answer is <math>\boxed{\textbf{(D) } 5}</math>.
 +
 +
~KingRavi
 +
 +
==Solution 2==
 +
 +
From the first equation, we can express <math>y</math> in terms of <math>x</math>:
  
 +
<math>3y = 9 - x^2</math> which is <math> y = (9 - x^2)/3</math>
  
−
</asy>
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The second equation can be rewritten as:
 +
 
 +
<math>|x| + |y| - 4 = \pm 1</math>.
 +
 
 +
This gives us two scenarios to examine:
 +
 
 +
1. <math>|x| + |y| = 5</math>
 +
 
 +
2. <math>|x| + |y| = 3</math>
 +
 
 +
Case 1:
 +
 
 +
Substituting <math>y</math>,
 +
 
 +
<math>|x| + |(9 - x^2)/3| = 5</math>.
 +
 
 +
First, consider the case when <math>9 - x^2 \geq 0</math>. Then,
 +
 
 +
<math>|y| = (9 - x^2)/3</math> which is <math>|x| + (9 - x^2)/3 = 5</math>.
 +
 
 +
Multiplying by 3, we get
 +
 
 +
<math>3|x| + 9 - x^2 = 15</math> which is <math>3|x| - x^2 = 6</math> which is <math>x^2 - 3|x| + 6 = 0</math>.
 +
 
 +
However, the discriminant of this quadratic is <math>(-3)^2 - 4 * 1 * 6 = 9 - 24 = -15</math>, which indicates there are no real solutions in this scenario.
 +
 
 +
Now, we can consider when <math>9 - x^2 < 0</math>
 +
 
 +
<math>|y| = -(9 - x^2)/3 = (x^2 - 9)/3</math> which is <math>|x| + (x^2 - 9)/3 = 5</math>.
 +
 
 +
Multiplying by 3, we get
 +
 
 +
<math>3|x| + x^2 - 9 = 15</math> which is <math>x^2 + 3|x| - 24 = 0</math>.
 +
 
 +
Now, let <math>u = |x|</math>, which gives <math>u^2 + 3u - 24 = 0</math>. When we calculate the discriminant, we get <math>3^2 - 4 * 1 * (-24) = 9 + 96 = 105</math>. So, the roots are <math>u = (-3 \pm \sqrt(105))/2</math>.
 +
 
 +
Both roots give positive values for <math>u</math>, resulting in two values of <math>x</math> for each root (one positive and one negative).
 +
 
 +
Case 2: \( |x| + |y| = 3 \)
 +
 
 +
Substituting <math>y</math>:
 +
 
 +
<math>|x| +  |(9 - x^2)/3| = 3</math>.
 +
 
 +
If <math>9 - x^2 \geq 0</math>, then <math>|y| = (9 - x^2)/3</math> which is |<math>x| + (9 - x^2)/3 = 3</math>.
 +
 
 +
Multiplying by 3, we get
 +
 
 +
<math>3|x| + 9 - x^2 = 9</math> which is <math>3|x| = x^2</math> which is <math>x^2 - 3|x| = 0</math>.
 +
 
 +
Thus, <math>|x|(|x| - 3) = 0</math>, leading to <math>x = 0</math> or <math>x = 3</math> (both giving corresponding <math>y</math> values).
 +
 
 +
If <math>9 - x^2 < 0</math>, then <math>|y| = (x^2 - 9)/3</math> which is <math>|x| + (x^2 - 9)/3 = 3</math>.
 +
 
 +
When we multiply through, we get <math>3|x| + x^2 - 9 = 9</math> which is <math>x^2 + 3|x| - 18 = 0</math>.
 +
 
 +
The discriminant here is <math>3^2 - 4 * 1 * (-18) = 9 + 72 = 81</math>.
 +
 
 +
This gives two more real roots for <math>u = |x|</math>.
 +
 
 +
Now,
 +
 
 +
- Case 1 contributes 2 solutions
 +
 
 +
- Case 2 contributes 1 solution from <math>x = 0</math> and <math>x = 3</math>, and 2 solutions from the second sub-case
 +
 
 +
Thus, counting all solutions gives us a total of 5 unique ordered pairs, and the answer is <math>\boxed{\textbf{(D) } 5}</math>.
 +
 
 +
~goofytaipan
  
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We see from the graph that there are 5 intersections, so the answer is <math>\boxed{D}</math>
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==Video Solution ==
 +
https://youtu.be/yASY-XL9vtI
  
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==Solution 2 (Unrigorous but pretty standard)==
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~Education, the Study of Everything
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We can manipulate the first equation to get <math>y = -\frac{x^{2}}{3} + 3</math>. From the second equation, we have that <math>|x|+|y|-4 = 1</math> or <math>|x|+|y|-4 = -1</math>. We will consider each case separately.
 
  
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If <math>|x|+|y|-4 = 1</math>, then <math>|x|+|y| = 5</math>. The graph of this is a square with vertices <math>(5,0)</math>, <math>(-5,0)</math>, <math>(0,5)</math> and <math>(0,-5)</math>. The parabola from the first equation is downwards facing, and its vertex is inside this square; the parabola will clearly intersect the square twice. Therefore, this case gives us <math>\underline{2}</math> solutions.
+
==Video Solution==
 +
https://youtu.be/zq3UPu4nwsE?t=974
  
−
If <math>|x|+|y|-4 = -1</math>, then <math>|x|+|y| = 3</math>. The graph of this is a square with vertices <math>(3,0)</math>, <math>(-3,0)</math>, <math>(0,3)</math> and <math>(0,-3)</math>. The vertex of the parabola from the first equation is on one of the corners of this square (in particular, <math>(0,3)</math>). Also, at <math>y = 0</math>, the parabola has <math>x</math> intercepts of <math>\pm 3</math>; the square passes through both of those points. If we continue to move down, the square narrows in, while the parabola continues to expand. Therefore, these are our only <math>3</math> intersection points in this case: <math>(0,3)</math>, <math>(3,0)</math> and <math>(-3,0)</math>. This case gives us <math>\underline{3}</math> solutions.
+
==Video Solution by WhyMath==
 +
https://youtu.be/5SVmxNrZUbY
  
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Adding these two cases together, we get our final answer of <math>\boxed{\textbf{(D) } 5}</math>.
+
~savannahsolver
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=A|num-b=13|num-a=15}}
 
{{AMC10 box|year=2021 Fall|ab=A|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 20:54, 2 November 2024

Problem

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \begin{align*} x^2+3y&=9 \\ (|x|+|y|-4)^2 &= 1 \end{align*} $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 7$

Solution 1 (Graphing)

The second equation is $(|x|+|y| - 4)^2 = 1$. We know that the graph of $|x| + |y|$ is a very simple diamond shape, so let's see if we can reduce this equation to that form: \[(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5\}.\] We now have two separate graphs for this equation and one graph for the first equation, so let's put it on the coordinate plane: [asy] Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 1.0,0.5)); yaxis(-8,8,Ticks(f, 1.0,0.5)); real f(real x) { return 3-x; } draw(graph(f,0,3)); real f(real x) { return 3+x; } draw(graph(f,0,-3)); real f(real x) { return x-3; } draw(graph(f,0,3)); real f(real x) { return -x-3; } draw(graph(f,0,-3)); real f(real x) { return 5-x; } draw(graph(f,0,5)); real f(real x) { return 5+x; } draw(graph(f,0,-5)); real f(real x) { return x-5; } draw(graph(f,0,5)); real f(real x) { return -x-5; } draw(graph(f,0,-5)); real f(real x) { return 3-x; } draw(graph(f,0,3)); real f(real x) { return 3+x; } draw(graph(f,0,-3)); real f(real x) { return x-3; } draw(graph(f,0,3)); real f(real x) { return (-x^2)/3+3; } draw(graph(f,-5,5)); [/asy] We see from the graph that there are $5$ intersections, so the answer is $\boxed{\textbf{(D) } 5}$.

~KingRavi

Solution 2

From the first equation, we can express $y$ in terms of $x$:

$3y = 9 - x^2$ which is $y = (9 - x^2)/3$

The second equation can be rewritten as:

$|x| + |y| - 4 = \pm 1$.

This gives us two scenarios to examine:

1. $|x| + |y| = 5$

2. $|x| + |y| = 3$

Case 1:

Substituting $y$,

$|x| + |(9 - x^2)/3| = 5$.

First, consider the case when $9 - x^2 \geq 0$. Then,

$|y| = (9 - x^2)/3$ which is $|x| + (9 - x^2)/3 = 5$.

Multiplying by 3, we get

$3|x| + 9 - x^2 = 15$ which is $3|x| - x^2 = 6$ which is $x^2 - 3|x| + 6 = 0$.

However, the discriminant of this quadratic is $(-3)^2 - 4 * 1 * 6 = 9 - 24 = -15$, which indicates there are no real solutions in this scenario.

Now, we can consider when $9 - x^2 < 0$

$|y| = -(9 - x^2)/3 = (x^2 - 9)/3$ which is $|x| + (x^2 - 9)/3 = 5$.

Multiplying by 3, we get

$3|x| + x^2 - 9 = 15$ which is $x^2 + 3|x| - 24 = 0$.

Now, let $u = |x|$, which gives $u^2 + 3u - 24 = 0$. When we calculate the discriminant, we get $3^2 - 4 * 1 * (-24) = 9 + 96 = 105$. So, the roots are $u = (-3 \pm \sqrt(105))/2$.

Both roots give positive values for $u$, resulting in two values of $x$ for each root (one positive and one negative).

Case 2: \( |x| + |y| = 3 \)

Substituting $y$:

$|x| + |(9 - x^2)/3| = 3$.

If $9 - x^2 \geq 0$, then $|y| = (9 - x^2)/3$ which is |$x| + (9 - x^2)/3 = 3$.

Multiplying by 3, we get

$3|x| + 9 - x^2 = 9$ which is $3|x| = x^2$ which is $x^2 - 3|x| = 0$.

Thus, $|x|(|x| - 3) = 0$, leading to $x = 0$ or $x = 3$ (both giving corresponding $y$ values).

If $9 - x^2 < 0$, then $|y| = (x^2 - 9)/3$ which is $|x| + (x^2 - 9)/3 = 3$.

When we multiply through, we get $3|x| + x^2 - 9 = 9$ which is $x^2 + 3|x| - 18 = 0$.

The discriminant here is $3^2 - 4 * 1 * (-18) = 9 + 72 = 81$.

This gives two more real roots for $u = |x|$.

Now,

- Case 1 contributes 2 solutions

- Case 2 contributes 1 solution from $x = 0$ and $x = 3$, and 2 solutions from the second sub-case

Thus, counting all solutions gives us a total of 5 unique ordered pairs, and the answer is $\boxed{\textbf{(D) } 5}$.

~goofytaipan

Video Solution

https://youtu.be/yASY-XL9vtI

~Education, the Study of Everything

Video Solution

https://youtu.be/zq3UPu4nwsE?t=974

Video Solution by WhyMath

https://youtu.be/5SVmxNrZUbY

~savannahsolver

See Also

2021 Fall AMC 10A (Problems • Answer Key • Resources)
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Problem 13 		Followed by
Problem 15
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