2021 Fall AMC 10A Problems/Problem 14

Revision as of 20:16, 23 November 2021 by Kingravi (talk | contribs) (Solution (Graphing))

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.

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

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 x+y-3;  }  draw(graph(f,-2,2));   [/asy]

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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