2021 Fall AMC 10A Problems/Problem 14

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:


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));


 (Error making remote request. Unknown error_msg)

2021 Fall AMC 10A (Problems • Answer Key • Resources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Page

Toolbox

Search

2021 Fall AMC 10A Problems/Problem 14

Problem

Solution (Graphing)

See Also