1987 AMC 12 Problema 15

Question

Si

(x,y)

es una solución del sistema

\begin{aligned} xy&=6,\\ x^2y+xy^2+x+y&=63, \end{aligned}

halla

x^2+y^2.

If

(x,y)

is a solution to the system

\begin{aligned} xy&=6,\\ x^2y+xy^2+x+y&=63, \end{aligned}

find

x^2+y^2.

13

\frac{1173}{32}

55

69

81

Factoriza la segunda ecuación usando

x+y

Factor the second equation using

x+y

Usa

x^2+y^2=(x+y)^2-2xy

Use

x^2+y^2=(x+y)^2-2xy

La segunda ecuación se convierte en

\begin{aligned} xy(x+y)&+(x+y)\\ &=(xy+1)(x+y)\\ &=63. \end{aligned}

Como

xy=6,

obtenemos

x+y=9.

Por tanto,

\begin{aligned} x^2+y^2 &=(x+y)^2-2xy\\ &=81-12=69. \end{aligned}

Por lo tanto, la respuesta correcta es D.

The second equation becomes

\begin{aligned} xy(x+y)&+(x+y)\\ &=(xy+1)(x+y)\\ &=63. \end{aligned}

Since

xy=6,

we get

x+y=9.

Therefore

\begin{aligned} x^2+y^2 &=(x+y)^2-2xy\\ &=81-12=69. \end{aligned}

Thus the correct answer is D.

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