The length of a flag is 0.3 foot less than twice its width. if the perimeter is 14.4 feet longer than the width, find the dimensions of the flag.

  The dimensions are:  5.7  ft.  by  3 ft.  .
 
(or, write as:  The length is 5.7 ft.   The width is 3 ft.).   {Note:  Do not forget to include the "units", which are "feet", or "ft.", in this case!}.

Explanation:

Consider a flag to be a "rectangle".  We are asked to find the dimensions; which are the "length" and the "width".  Let "l" represent the length; and "w" represent the "width".   The "area", or "A", of a rectangle = the length multiplied by the width;  or:   "A = l * w " .   

So, given: "the length of a flag is 0.3 foot less than twice its width" ;

    We express this as:  " l = 2w - 0.3 " .

Now, note that the Perimeter, "P", of the rectangle:

 " P = 2l + 2w "  ; 

We are given:  "the perimeter is 14.4 feet longer than the width" ;

We express this as:  "P = 14.4 + w " ;

so:  P = 2l + 2w  = 14.4 + w ;

then:   2l + 2w = 14.4 + w . 

Since:  " l = 2w - 0.3" ;  we can substitute this value, "(2w - 0.3)" ,                                                for the "l" in the above equation; and solve for "w" ;

                    →   2l + 2w = 14.4 + w  ;

                    →   2(2w - 0.3) + 2w = 14.4 + w ; 

Now, subtract "w" from EACH SIDE of the equation:

     2(2w - 0.3) + 2w - w = 14.4 + w - w ;

to get:  2(2w - 0.3) + w = 14.4;

Note the distributive property of multiplication:

a(b+c) = ab + ac ; AND:
a(b -c) = ab - ac ;

So;   2(2w - 0.3) = (2*2w) - (2* 0.3) = 4w - 0.6 ; 

Rewrite the equation, substituting:  "4w - 0.6" in lieu of:  "2(2w - 0.3)" ;

→ 2(2w - 0.3) + w = 14.4 ;  

→ Rewrite as:  4w - 0.6 + w = 14.4 ;
 
 Now, combine the "like terms" that appear on the left-hand side of the equation:   +4w  + w = 5w ;  and rewrite the equation:

   →  5w - 0.6 = 14.4 ;

   →  Now, add "0.6" to EACH side of the equation:

             →  5w - 0.6 + 0.6 = 14.4 + 0.6 ; 

      to get:   5w = 15 ; 

Now, divide EACH side of the equation by "5" ;  to isolate "w" on one side of the equation; and to solve for "w" (the width, which is one of the dimensions); 

         →  5w / 5  = 15/ 5 ;

         →  w = 3 ;


Now, we need to find the length, "l".

Since, " l = 2w - 0.3 ;  we can substitute our known value of "w", which is "3", into the equation, and solve for "l" ;

         →  l = (2*3) - 0.3 = 6 - 0.3 = 5.7 ;
       
         →  l = 5.7 .

Now, to check our work, does:

2l + 2w = 14.4 + w  ?  

Let us plug our solved values, "5.7" for "l", and "3" for "w"; so see if the equation holds true.

Let us start with the right side of the equation:

14.4 + w ;   14.4 + w = 14.4 + 3 = 17.4. 

Now, let us consider the left side of of the equation:

2l + 2w = (2*5.7) + (2*3) = 11.4 + 6 = 17.4 .

17.4 =? 17.4 ?  Yes!.

So, the answer is:  The dimensions are:  5.7  ft.  by  3 ft.  .

(or, write as:  The length is 5.7 ft.   The width is 3 ft.).

{Note:  Do not forget to include the "units", which are "feet", or "ft.", in this case!}.

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