Answer
Length: $13\; yards$.
Width: $4\; yards$.
Work Step by Step
Let the width of the rectanglebe $W=x$.
From the question the length of the rectangle is
$L=3x+1$.
Formula for the area of the rectangle is $A=W\cdot L$. (1)
We are given that $A=52\;square \;yards$.
Plug $A=52$, $W=x$ and $L=3x+1$ into (1).
$52=x(3x+1)$
Clear the parentheses.
$52=3x^2+x$
Subtract $52$ from both sides.
$52-52=3x^2+x-52$
Simplify.
$0=3x^2+x-52$
Rewrite the middle term $x$ as $13x-12x$.
$0=3x^2+13x-12x-52$
Group the terms.
$0=(3x^2+13x)+(-12x-52)$
Factor each term.
$0=x(3x+13)-4(3x+13)$
Factor out $(3x+13)$.
$0=(3x+13)(x-4)$
Set each factor equal to zero.
$3x+13=0$ or $x-4=0$
Isolate $x$.
$x=-\frac{13}{3}$ or $x=4$
Take positive value because $x$ represents a dimension.
Width of the rectangle is $4 \; yards$
Length $L=3(4)+1$
$L=12+1$
$L=13\; yards$.
