Difference between revisions of "2021 Fall AMC 10A Problems/Problem 14"
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Revision as of 19:20, 26 September 2023
Contents
Problem
How many ordered pairs 
of real numbers satisfy the following system of equations?
Solution 1 (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: ![\[(|x|+|y| - 4)^2 = 1 \implies |x|+|y| - 4 = \pm 1 \implies |x|+|y| = \{3,5\}.\]](http://latex.artofproblemsolving.com/4/e/a/4eae12e93fc1db2f8b8038aebdd206a7a6b5c323.png)
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]](http://latex.artofproblemsolving.com/5/1/b/51bfd5ebea99685462e153dbecd9a825da8b2056.png)
We see from the graph that there are 
intersections, so the answer is 
.
~KingRavi
Video Solution
~Education, the Study of Everything
Video Solution
https://youtu.be/zq3UPu4nwsE?t=974
Video Solution by WhyMath
~savannahsolver
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
