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Difference between revisions of "1987 AHSME Problems/Problem 15"

(Problem)
m (Fixed formatting in the problem statement)
 
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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 \qquad \text{and} \qquad x^2y+xy^2+x+y=63</math>,
+
<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>.
  

Latest revision as of 11:36, 31 March 2018

Problem

If $(x, y)$ is a solution to the system $xy=6$ and $x^2y+xy^2+x+y=63$, find $x^2+y^2$.

$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ \frac{1173}{32} \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 69 \qquad \textbf{(E)}\ 81$

Solution

First note that $x^2y+xy^2+x+y= (xy+1)(x+y)$. Substituting $6$ for $xy$ gives $7(x+y)= 63$, giving a result of $x+y=9$. Squaring this equation and subtracting by $12$, gives us $x^2+y^2= \boxed{69}$

See also

1987 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AHSME Problems and Solutions

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