Question 194145
Let P = perimeter, L=length, and W=width



Note: the lower case letter "l" looks very similar to the number 1. So I'm using an uppercase L. So I figured to use uppercase for all the variables.



Now let's get the translations down:


"The perimeter of a rectangle is six times the width" translates to {{{P=6W}}}


Also, "If the length were increased by 4 inches and the width were increased by 7 inches, then the perimeter would be 160 inches" translates to {{{160=2(L+4)+2(W+7)}}}


Finally, this is not explicitly stated, but the formula for the perimeter of any rectangle is {{{P = 2L + 2W}}}



{{{P = 2L + 2W}}} Start with the given perimeter formula



{{{6W = 2L + 2W}}} Plug in {{{P=6W}}}



{{{6W - 2W = 2L }}} Subtract 2W from both sides



{{{4W = 2L }}} Combine like terms.



{{{2L = 4W }}} Rearrange the equation



{{{L = (4W)/2 }}} Divide both sides by 2 to isolate L



{{{L = 2W }}} Reduce



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{{{160=2(L+4)+2(W+7)}}} Move onto the second equation



{{{160=2L+8+2W+14}}} Distribute 



{{{160=2L+2W+22}}} Combine like terms.



{{{160=2(2W)+2W+22}}} Plug in {{{L = 2W }}}



{{{160=4W+2W+22}}} Multiply



{{{160=6W+22}}} Combine like terms.



{{{160-22=6W}}} Subtract 22 from both sides.



{{{138=6W}}} Combine like terms.



{{{138/6=W}}} Divide both sides by 6 to isolate W



{{{23=W}}} Reduce



So the width is {{{W=23}}}



{{{L=2W}}} Go back to the previously isolated formula



{{{L=2(23)}}} Plug in {{{W=23}}}



{{{L=46}}} Multiply



So the length is {{{L=46}}}



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Answer:


So the width is 23 inches and the length is 46 inches.