Verified answer

The original figure was a square with equal sizes to each side, by definition... call each original side's length "x" ... the area of the original square was x*x.

x*x - 21 = the area of the new rectangle

(x+1)*(x-3) = the area of the new rectangle also

thus:

x*x - 21 = (x+1)*(x-3)

x*x - 21 = x*x - 3x + x - 3

3x - x = 18

2x = 18

x = 9

The area of the original square was 9*9 = 81 square centimeters

The area of the new rectangle is (9+1) * (9-3) = 10 * 6 = 60 cm^2

The difference between the areas is 81 - 60 = 21 centimeters squared, as required.

Let s = length of a side of a square

Let L_Rectangle = length of the new rectangle

Let W_Rectangle = width of the new rectangle

If you are having trouble translating words into expressions, break up what you're given into smaller pieces first.

Given: one side of a square

Means: s

Given: is increased by 1 cm

Means: + 1

Put them back together.

Given: One side of a square is increased by 1 cm

Means: L_Rectangle = s + 1

Given: An adjacent side

Implied: You remember that a square's sides are all the same length

Means: s

Given: is decreased by 3 cm

Means: - 3

Put them back together.

Given: An adjacent side is decreased by 3 cm.

Means: W_Rectangle = s - 3

Given: The area of the resulting rectangle

Implied: You know the area formula for a rectangle: A = L * W

Means: Area_Rectangle = L_Rectangle * W_Rectangle

Given: is 21 cm^2 less than

Means: -21 +

Given: that of the square

Implied: You know the area formula for a square: A = s^2

Means: s^2

Put them back together.

Given: The area of the resulting rectangle is 21cm^2 less than that of the square.

Means: L_Rectangle * W_Rectangle = s^2 - 21

Your system of 3 equations is:

L_Rectangle = s + 1

W_Rectangle = s - 3

L_Rectangle * W_Rectangle = s^2 - 21

Substitute L_Rectangle with s + 1 in the third equation.

L_Rectangle * W_Rectangle = s^2 - 21

(s + 1) * W_Rectangle = s^2 - 21

Substitute W_Rectangle with s - 3.

(s + 1) * (s - 3) = s^2 - 21

(s + 1)(s - 3) = s^2 - 21

Expand.

s(s) + s(-3) + 1(s) + 1(-3) = s^2 - 21

s^2 - 3s + s - 3 = s^2 - 21

s^2 - 2s - 3 = s^2 - 21

Subtract s^2 from both sides.

s^2 - 2s - 3 - s^2 = s^2 - 21 - s^2

-2s - 3 = -21

Add 3 to both sides.

-2s - 3 + 3 = -21 + 3

-2s = -18

Divide both sides by -2.

-2s / -2 = -18 / -2

s = 9

ANSWER: The square's sides are 9 cm each.

CHECK:

L_Rectangle = s + 1

L_Rectangle = 9 + 1

L_Rectangle = 10

W_Rectangle = s - 3

W_Rectangle = 9 - 3

W_Rectangle = 6

L_Rectangle * W_Rectangle = s^2 - 21

10 * 6 = 9^2 - 21?

60 = 81 - 21?

60 = 60?

TRUE

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