Difference between revisions of "2021 Fall AMC 10A Problems/Problem 14"
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==Solution (Graphing)== | ==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>. | ||
| + | |||
| + | 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] | ||
==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 19:42, 23 November 2021
Problem
How many ordered pairs 
of real numbers satisfy the following system of equations?
![\[x^2+3y=9\]](http://latex.artofproblemsolving.com/c/0/9/c09aed7a84906eb65f1fa163a7a7d92168877583.png)
![\[(|x|+|y|-4)^2 = 1\]](http://latex.artofproblemsolving.com/4/d/d/4dd5d2dee801931df0406925786a98be62e43cc2.png)
Solution (Graphing)
The second equation is 
. We know that the graph of 
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]
See Also
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
