Answer
4 and 6
Work Step by Step
We write the system:
$\begin{cases}
x+y=10\\
xy=24
\end{cases}$
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 $y$:
$\begin{cases}
x=10-y\\
(10-y)y=24
\end{cases}$
$10y-y^2=24$
$10y-y^2-24=0$
$y^2-10y+24=0$
$y^2-4y-6y+24=0$
$y(y-4)-6(y-4)=0$
$(y-4)(y-6)=0$
$y-4=0\Rightarrow y_1=4$
$y-6=0\Rightarrow y_2=6$
Substitute each of the values of $y$ in the expression of $x$ to determine $x$:
$x=10-y$
$y_1=4\Rightarrow x_1=10-4=6$
$y_2=6\Rightarrow x_2=10-6=4$
The system's solutions are:
$(6,4), (4,6)$
The numbers are 4 and 6.
