Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6"
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If <math>z=0</math>, then <math>x+y = 0</math> and <math>xy + 2y = 0</math>. We can rearrange and solve: | If <math>z=0</math>, then <math>x+y = 0</math> and <math>xy + 2y = 0</math>. We can rearrange and solve: | ||
− | <math>xy + 2y = 2x + 2y Rightarrow xy = 2x \Rightarrow x=0, y =2</math>. | + | <math>xy + 2y = 2x + 2y \Rightarrow xy = 2x \Rightarrow x=0, y =2</math>. |
This results in solutions: <math>(x,y,z) = (0,0,0), (-2,2,0)</math>. | This results in solutions: <math>(x,y,z) = (0,0,0), (-2,2,0)</math>. |
Revision as of 20:56, 25 July 2016
Problem
How many triples of rational numbers satisfy the following system of equations?
Solution
If , then
and
. We can rearrange and solve:
.
This results in solutions: .
If ,
,
and
. We can set up a 3rd degree polynomial
with roots
.
.
See also
2014 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |