TITLE: what would be the way to solve a system of equations like this one?
QUESTION [9 upvotes]: Solve:
$xy=-30$
$x+y=13$
{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?

REPLY [4 votes]: The other answers here show the strategies of:

$1$) Substitution (the most general). See Kim Jong Un's answer on this question.

$2)$ Solving a quadratic equation that has the roots $x,y$ (this uses the Vieta's formulas). See Sami Ben Romdhane's answer.

$3)$ Using the fact that $(x+y)^2-4xy=(x-y)^2$ to have the numeric value of $x-y$. Then $(x+y)-(x-y)=2y$, an easy way to get the value of $y$. See André Nicolas' answer.

I'm posting another method that works great for this particular problem.
Add the equations in a specific way to get:$$\begin{align}xy+2(x+y)&=-30+2\cdot 13=-4\\\iff (x+2)(y+2)&=0\end{align}$$
Thus either $x=-2$ or $y=-2$, which give the solutions $(-2,15)$ and $(15,-2)$, respectively.