Answer
Length: 11 feet
Width: 7 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 pf the rectangle:
$\begin{cases}
2l+2w=36\\
lw=77
\end{cases}$
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 $l$:
$\begin{cases}
w=\dfrac{36-2l}{2}\\
lw=77
\end{cases}$
$\begin{cases}
w=18-l\\
lw=77
\end{cases}$
$l(18-l)=77$
$18l-l^2=77$
$l^2-18l+77=0$
$l^2-7l-11l+77=0$
$l(l-7)-11(l-7)=0$
$(l-7)(l-11)=0$
$l-7=0\Rightarrow l_1=7$
$l-11=0\Rightarrow l_2=11$
Substitute each of the values of $l$ in Equation 2 to determine $w$:
$lw=77$
$l_1=7\Rightarrow 7w=77\Rightarrow w_1=\dfrac{77}{7}\Rightarrow w_1=11$
$l_2=11\Rightarrow 11w=77\Rightarrow w_2=\dfrac{77}{11}\Rightarrow w_2=7$
The system's solutions are:
$(7,11),(11,7)$
As $l\geq w$, the solution is:
$l=11$ ft
$w=7$ ft
$(-1,-1), (1,-1), (-1,1),(1,1)$
