Answer
8 and 12
Work Step by Step
We write the system:
$\begin{cases}
x+y=20\\
xy=96
\end{cases}$
We will use the substitution method. Solve Equation 1 for $x$ and substitute the expression of $x$ in Equation 2 to eliminate $x$ and determine $y$:
$\begin{cases}
x=20-y\\
(20-y)y=96
\end{cases}$
$20y-y^2=96$
$20y-y^2-96=0$
$y^2-20y+96=0$
$y^2-8y-12y+96=0$
$y(y-8)-12(y-8)=0$
$(y-8)(y-12)=0$
$y-8=0\Rightarrow y_1=8$
$y-12=0\Rightarrow y_2=12$
Substitute each of the values of $y$ in the expression of $x$ to determine $x$:
$x=20-y$
$y_1=8\Rightarrow x_1=20-8=12$
$y_2=12\Rightarrow x_2=20-12=8$
The system's solutions are:
$(12,8), (8,12)$
The numbers are 8 and 12.