a house has two rooms of equal area. one room is square and the other room a rectangle, four feet narrower and 5ft longer than the square one. find the area of each room?

let x = dimensions of the square.
Rectangle then is x-4 and x+5
Set areas equal.
area of square is x2
area of rectangle is (x-4)*(x+5).

(x-4)*(x+5)=x2
solve for x which will be sides of the square. Square that to obtain the area.
Post your work if you get stuck.

To find the area of each room, let's follow the steps you provided.

1. Let x represent the dimensions (side length) of the square room.
2. The rectangle room is described as being four feet narrower and five feet longer than the square room. So, the dimensions of the rectangle are (x-4) and (x+5).
3. The area of the square room is given by x^2 (the side length squared).
4. The area of the rectangle room is given by (x-4)*(x+5) (length multiplied by width).

Now, we'll set the two areas equal to each other:

x^2 = (x-4)*(x+5)

To solve for x, we'll distribute and simplify the right side:

x^2 = (x^2 + x*5 - x*4 - 4*5)
x^2 = (x^2 + 5x - 4x - 20)
x^2 = (x^2 + x - 20)

Next, bring all terms to one side to set the equation equal to zero:

x^2 - x^2 - x + 20 = 0

Simplifying further:

-x + 20 = 0
-x = -20
x = 20

So, the side length of the square room (x) is 20 feet.

To find the area of the square room, square the side length: 20^2 = 400 square feet.

As for the rectangle room, its dimensions are (20-4) and (20+5), which simplifies to 16 and 25, respectively.

To find the area of the rectangle room, multiply the length by the width: 16 * 25 = 400 square feet.

Hence, the area of both rooms is 400 square feet.