Track / Stage 5 / 85 of 400 #685 of 1964

Problem 685

AIME late

Algebra Difficulty 5.1 Find the answer

Determine all pairs $(x ; y)$ of real numbers $x, y$ that satisfy the following system of equations (1), (2), (3):

$\begin{aligned} x + xy + xy^2 & = -21 \\ y + xy + x^2 y & = 14 \\ x + y & = -1 \end{aligned}$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

Adding (1) and (2) results in the new equation

$(x+y)+2 x y+x y(x+y)=-7$

from which $x y=-6$ is immediately obtained using (3).

By substituting this into (1), one obtains the additional linear equation

$x-6 y=-15$

which, in conjunction with (3), ultimately leads to the unique solution $x=-3, y=2$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.