Question

The length of a picture frame is 2 inches more than the width. The perimeter of the frame is 36 inches. What is the length of the frame?

You will earn 2 points for writing the correct system of equations for this word problem. Remember, two equations must be given for the system of equations. You will earn 3 points for solving it correctly and answering the question in a complete sentence.

Answer 1

Answer in one complete sentence:
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   "The length of the frame is ten (10) inches."
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Explanation:
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Perimeter = P = 36 in.
Length = L = 2 +2
Width = w .
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These are the two equations for the system of equations:

P = 2L + 2w = 36 in.
L = 2 + w
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Solve for "L" (length) of the frame.
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In the equation:
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P = 2L + 2w = 36.

{Note: standard knowledge; and/or look up: 
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Formula for the Perimeter of a rectangle:  P = 2L + 2w} ;
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We are given:  P = 36 in.
 
Rewrite, substituting "2+w" for "L" ;

P = 2(2+w) + 2w = 36 ;
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Note:  2(2+w) = 2*2 + 2w = 4 + 2w .

Now, Rewrite the equation, substituting "(4 + 2w) for "2(2+w)" ;
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P = (4 + 2w)  + 2w = 36 ;

P =  4 + 2w + 2w = 36 ;

Combine the "like terms" ;
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  + 2w + 2w = 4w ;
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   P = 4 + 4w = 36 ;  
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↔   4 + 4w = 36 ;
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Subtract "4" from EACH SIDE of the equation:
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  →  4 + 4w − 4 = 36 − 4 ;
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 to get:
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            →  4w  =  32 ;
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            → Divide EACH SIDE of the equation by "4" ; to isolate "w" on one side of the equation; and to solve for "w" ;  
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            →  4w / 4 = 32 / 4 ;

               →  w = 8 .
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Now, to solve for: "L" (the length).
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    Length:  L = 2 + w ; 

                 →  L = 2 + 8 = 10 inches.
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Answer in one complete sentence:
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   "The length of the frame is ten (10) inches."
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