Difference between revisions of "1987 AHSME Problems/Problem 15"
(Created page with "==Problem== If <math>(x, y)</math> is a solution to the system <math>xy=6 \qquad \text{and} \qquad x^2y+xy^2+x+y=63</math>, find <math>x^2+y^2</math>. <math>\textbf{(A)}\ 13 \q...") |
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If <math>(x, y)</math> is a solution to the system | If <math>(x, y)</math> is a solution to the system | ||
| − | <math>xy=6 | + | <math>xy=6</math> and <math>x^2y+xy^2+x+y=63</math>, |
find <math>x^2+y^2</math>. | find <math>x^2+y^2</math>. | ||
| Line 11: | Line 11: | ||
\textbf{(E)}\ 81 </math> | \textbf{(E)}\ 81 </math> | ||
| + | ==Solution== | ||
| + | First note that <math>x^2y+xy^2+x+y= (xy+1)(x+y)</math>. Substituting <math>6</math> for <math>xy</math> gives <math>7(x+y)= 63</math>, giving a result of <math>x+y=9</math>. Squaring this equation and subtracting by <math>12</math>, gives us <math>x^2+y^2= \boxed{69}</math> | ||
== See also == | == See also == | ||
Latest revision as of 11:36, 31 March 2018
Problem
If 
is a solution to the system
and 
,
find 
.
Solution
First note that 
. Substituting 
for 
gives 
, giving a result of 
. Squaring this equation and subtracting by 
, gives us 
See also
| 1987 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
