Answer
Length: 12 feet
Width: 8 feet
Work Step by Step
Let's note:
$l$=the length of the rectangle
$w$=the width of the rectangle
We can write the system, using the perimeter and the area of the rectangle:
$\begin{cases}
2l+2w=40\\
lw=96
\end{cases}$
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 $l$:
$\begin{cases}
w=\dfrac{40-2l}{2}\\
lw=96
\end{cases}$
$\begin{cases}
w=20-l\\
lw=96
\end{cases}$
$l(20-l)=96$
$20l-l^2=96$
$l^2-20l+96=0$
$l^2-8l-12l+96=0$
$l(l-8)-12(l-8)=0$
$(l-8)(l-12)=0$
$l-8=0\Rightarrow l_1=8$
$l-12=0\Rightarrow l_2=12$
Substitute each of the values of $l$ in Equation 2 to determine $w$:
$lw=96$
$l_1=8\Rightarrow 8w=96\Rightarrow w_1=\dfrac{96}{8}\Rightarrow w_1=12$
$l_2=12\Rightarrow 12w=96\Rightarrow w_2=\dfrac{96}{12}\Rightarrow w_2=8$
The system's solutions are:
$(8,12),(12,8)$
As $l\geq w$, the solution is:
$l=12$ ft
$w=8$ ft
$(-1,-1), (1,-1), (-1,1),(1,1)$
