Final answer:
The side of the square is 7 meters, the rectangle's width is 14 meters, and its length is 11 meters. The area of both the square and the rectangle is 49 square meters, which is derived from the side length of the square.
Step-by-step explanation:
Solving for the Side of the Square and Rectangle Dimensions
Let the side of the square be x meters. The area of a square is calculated as the side length squared, so the area of the square is x².
According to the problem, the length of the rectangle is twice the side of the square minus three meters (2x - 3), and the width is the side of the square plus seven meters (x + 7). As the square and rectangle have the same area, we set up the equation x² = (2x - 3)(x + 7).
Expanding the right side of the equation, we have x² = 2x² + 14x - 3x - 21. Simplifying, this gives us x² = 2x² + 11x - 21. To solve for x, move all terms to one side: x² - 2x² - 11x + 21 = 0, or x² - 11x + 21 = 0. This is a quadratic equation and can be solved for x using the quadratic formula, factoring, or completing the square.
In this case, factoring gives us (x - 3)(x - 7) = 0, leading to two possible values for x: x = 3 meters or x = 7 meters. However, since (2x - 3) must be positive, x must be greater than 1.5, so we discard x = 3 and take x = 7 meters as the side of the square.
Subsequently, the width of the rectangle is x + 7 = 14 meters, and the length of the rectangle is 2x - 3 = 11 meters. The area of the rectangle is the same as the square, which is 49 square meters.