area of a rectangle is defined by the formula
A = l x w
and perimeter by the formula
P = 2l + 2w
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given that area is 84
and perimeter is 38, you can substitute it to both equations stated above
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you'll have
A = l x w
84 = l x w ----equation 1
P = 2l + 2w
38 = 2l + 2w ----equation 2
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divide equation2 by 2
(38 = 2l + 2w) ÷ 2
19 = l + w
l = 19 - w -----equation 3
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substitute equation 3 to equation 1
84 = l (w)
84 = (19-w)(w)
84 = 19w -w^2
transpose 19w - w^2 to the left side of the equation
w^2 - 19w + 84 = 0
factor
(w-12)(w-7) = 0
w-12=0
w=12
w-7=0
w=7
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if w=12
substitute to equation 3
l = 19 - w
l = 19 - 12
l = 7
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if w=7
l = 19-w
l = 19-7
l = 12
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but common knowledge will tell you that length must be of greater value than the width.
then you can say that
l = 12
w = 7
