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 need to multiply the length and width. Let's start by finding the dimensions of each rectangle.
Rectangle 1: Length = a, Width = b
Rectangle 2: Length = a - 3, Width = b + 2
Rectangle 3: Length = a + 3, Width = b - 1
The area of each rectangle is given by:
Rectangle 1: Area1 = Length * Width = a * b
Rectangle 2: Area2 = (Length - 3) * (Width + 2) = (a - 3) * (b + 2)
Rectangle 3: Area3 = (Length + 3) * (Width - 1) = (a + 3) * (b - 1)
Since all three rectangles have the same area, we can equate the areas:
Area1 = Area2 = Area3
Let's solve this equation to find the value of a and b.
a * b = (a - 3) * (b + 2) = (a + 3) * (b - 1)
Expanding the equations, we get:
ab = ab + 2a - 3b - 6 = ab - a + 3b - 3
Now, we can cancel out ab from both sides of the equation:
0 = 2a - 3b - 6 = -a + 3b - 3
Next, we can combine like terms:
2a - 3b - 6 = -a + 3b - 3
Adding a to both sides:
2a - 3b + a - 3b - 6 = 3b - 3b + a - 3
Combining like terms, we get:
3a - 6b - 6 = a - 3
Simplifying further:
3a - a = 6b - 3b + 6 - 3
2a = 3b + 3
Dividing by 2 on both sides:
a = (3b + 3) / 2
Now, we have the relationship between a and b.
To find the actual values of a and b, we need more information or another equation to solve the system of equations.