If the width of a rectangle is increased by 2 inches, the area is 140 square inches. If the legnth is increased by 1 inch, the area is 120 inches. Find the dimensions of the rectangle.

To find the dimensions of the rectangle, we can set up a system of two equations based on the given information.

Let's assume the width of the rectangle is 'w' inches and the length is 'l' inches.

According to the first condition, if the width is increased by 2 inches, the new width would be 'w + 2' inches, and the area would be 140 square inches:

Area = Length × Width
140 = l × (w + 2) ---- Equation 1

According to the second condition, if the length is increased by 1 inch, the new length would be 'l + 1' inches, and the area would be 120 square inches:

Area = Length × Width
120 = (l + 1) × w ---- Equation 2

Now we have a system of two equations:
140 = l × (w + 2)
120 = (l + 1) × w

To solve this system, we can use the method of substitution.

From Equation 2, we can solve for l in terms of w:
l = (120 / w) - 1

Now we substitute this expression for l in Equation 1:
140 = [(120 / w) - 1] × (w + 2)

Simplifying this equation will give us a quadratic equation, which we can solve for 'w' to find the width. Once we have the width, we can plug it back into Equation 2 to find the length.

Let's solve this step-by-step.

Expanding the equation:
140 = (120 / w - 1)w + 2(120 / w - 1)

Simplifying further:
140 = 120 - w + 240 / w - 2

Combining like terms:
140 = 118 - w + 240 / w

Rearranging the terms:
w - 240 / w = 118 - 140

w - 240 / w = -22

Multiplying through by w to eliminate the fraction:
w^2 - 240 = -22w

Moving all terms to one side to get a quadratic equation:
w^2 + 22w - 240 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula.

Factoring:
(w + 30)(w - 8) = 0

Setting each factor equal to zero:
w + 30 = 0 or w - 8 = 0

Solving for w:
w = -30 or w = 8

Since the width of the rectangle cannot be negative, we can ignore the solution w = -30. Therefore, w = 8 inches.

Now we can substitute this value of w back into Equation 2 to solve for the length:

120 = (l + 1) × 8
120 = 8l + 8
112 = 8l
l = 14

So, the dimensions of the rectangle are:
Width = 8 inches
Length = 14 inches

From the description above, we know that:

(W+2) x L = 140
(L+1) x W = 120

where W = the width and L is the length, so...

WL + 2L = 140
WL + W = 120

Subtract the second equation from the first, you get...

2L - W = 20, so
W = 2L - 20.

So feed that back into the first of the original two equations:

(2L - 18) x L = 140

2L² - 18L - 140 = 0
so L² - 9L - 70 = 0
factorize it: (L - 14)(L + 5) = 0

so L = 14 or L = -5. The latter is impossible, so L = 14.
But W = 2L - 20, so W = 8.

Check it:

(W+2) x L = 10 x 14 = 140 Correct.
(L+1) x W = 15 x 8 = 120 Correct.

Answer: W = 8; L = 14.