Answer
$(12,15), \left(\dfrac{15}{2},24\right)$
Work Step by Step
Let's note:
$l$=the length of the rectangle
$w$=the width of the rectangle
We build the system:
$\begin{cases}
2l+w=39\\
lw=180
\end{cases}$
We will use the substitution method. Solve Equation 1 for $w$ and substitute the expression of $w$ in Equation 2 to eliminate $w$ and determine $l$:
$\begin{cases}
w=39-2l\\
l(39-2l)=180
\end{cases}$
$l(39-2l)=180$
$39l-2l^2=180$
$2l^2-39l+180=0$
$2l^2-24l-15l+180=0$
$2l(l-12)-15(l-12)=0$
$(l-12)(2l-15)=0$
$l-12=0\Rightarrow l_1=12$
$2l-15=0\Rightarrow l_2=\dfrac{15}{2}$
Substitute each of the values of $l$ in the expression of $w$ to determine $w$:
$w=39-2l$
$l_1=12\Rightarrow w_1=39-2(12)=15$
$l_2=\dfrac{15}{2}\Rightarrow w_2=39-2\left(\dfrac{15}{2}\right)=24$
The system's solutions are:
$(12,15), \left(\dfrac{15}{2},24\right)$
