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

(Solution (Graphing))
(Solution (Graphing))
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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:
  
[asy]
+
<math>[asy]
  
 
Label f;  
 
Label f;  
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[/asy]
+
[/asy]</math>
  
 
==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}}

Revision as of 19:43, 23 November 2021

Problem

How many ordered pairs $(x,y)$ of real numbers satisfy the following system of equations? \[x^2+3y=9\] \[(|x|+|y|-4)^2 = 1\]

$\textbf{(A )} 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 7$


Solution (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.

$(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5}$ (Error compiling LaTeX. Unknown error_msg).

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, 2.0,0.5)); yaxis(-8,8,Ticks(f, 2.0,0.5))

real f(real x) { return |x| + |y| = 3; } draw(graph(f,-2,2));

real f(real x) { return |x| + |y| = 5; } draw(graph(f,-2,2));

real f(real x) { return x^2+3y = 9; } draw(graph(f,-2,2));



[/asy]$ (Error making remote request. Unknown error_msg)

See Also

2021 Fall AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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

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