Answer: 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!}. __________________________________________________
