Question 966128
Let:
W=width; L=length=W+7in; D=diagonal
The diagonal of a rectangle forms the hypotenuse of a right triangle with the length and width as legs, so:
.
{{{L^2+W^2=D^2}}} Substitute for L
{{{(W+7in)^2+W^2=(13in)^2}}}
{{{(W^2+14W+49)+W^2=169in^2
{{{2W^2+14W+49=169in^2}}} Subtract 169in^2 from each side.
{{{2W^2+14W-120=0}}} Divide each side by 2.
{{{W^2+7W-60=0}}} 
{{{(W+12)(W-5)=0}}}
{{{W+12=0}}} {{{or}}} {{{W-5=0}}}
{{{W=-12}}} {{{or}}} {{{W=5}}}
ANSWER 1: The width of the frame is 5 inches.
.
L=W+7in=5in+7in=12in 
ANSWER 2: The length of the frame is 12 inches.
.
CHECK: 
{{{L^2+W^2=D^2}}}
{{{(12in)^2+(5in)^2=(13in)^2}}}
{{{144in^2+25in^2=169}}}
{{{169in^2=169in^2}}}