Three rectangles have exactly the same area. The dimensions of each rectangle (as length and width) are a and b; a – 3 and b + 2; and a + 3 and b – 1. Find the area of the rectangles.

To find the area of each rectangle, we first need to find the dimensions of each rectangle. Given that all three rectangles have the same area, we can set up an equation using the formula for the area of a rectangle: Area = length × width.

Let's start by labeling the dimensions of the first rectangle as a and b.

For the first rectangle, the length is a and the width is b.

For the second rectangle, the given dimensions are a - 3 and b + 2.

For the third rectangle, the given dimensions are a + 3 and b - 1.

Now, we can set up equations based on the given information:

1) Area of the first rectangle = a × b ---- Equation (1)
2) Area of the second rectangle = (a - 3) × (b + 2) ---- Equation (2)
3) Area of the third rectangle = (a + 3) × (b - 1) ---- Equation (3)

Since all three rectangles have the same area, we can equate the three equations:

a × b = (a - 3) × (b + 2) = (a + 3) × (b - 1)

Now, let's solve this equation to find the value of a × b, which represents the area of each rectangle.

Expanding the equation:

a × b = (a - 3) × (b + 2) = (a + 3) × (b - 1)
a × b = ab - 3b + 2a - 6 = ab + 3b + 2a - 3

Simplifying the equation:

ab - 3b + 2a - 6 = ab + 3b + 2a - 3

The terms with ab and 2a cancel out:

-3b - 6 = 3b - 3

Combining like terms:

-3b - 6 - 3b = 3b - 3 - 3b
-6b - 6 = -3

Moving the constant term to the other side:

-6b = -3 + 6

Simplifying:

-6b = 3

Dividing both sides by -6:

b = -3/6
b = -1/2

Now that we have the value of b, we can substitute it back into one of the equations to find the value of a.

Using Equation (1), where the length is a and the width is b:

a × (-1/2) = a × b
-a/2 = a × (-1/2)

Since the left side is negative, both a/2 and a must be negative, or both a/2 and a must be positive. Let's assume they are both positive:

-a/2 = -a/2

Now, we can cross-multiply:

-2a = -a
-2a + a = 0
-a = 0

This implies that a = 0. However, a rectangle cannot have a dimension of 0, so let's consider the case where a and a/2 are both negative:

-a/2 = a/2

Cross-multiply:

-2a = 2a
-2a - 2a = 0
-4a = 0

Dividing both sides by -4:

a = 0

But again, a rectangle cannot have a dimension of 0. Therefore, there is no valid solution for the dimensions a and b that satisfy all three equations.

Hence, it is not possible to find the area of the rectangles with the given conditions.