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------------------------------------------------ Which system of equations below can be used to solve the following problem?

The area of a rectangular room is 156 square feet. If the length of one side of the room is 13 feet, what is the perimeter of the room?

A. [tex] A = 13w [/tex], [tex] 156 = 2(\theta + w) [/tex]

B. [tex] 156 = 13w [/tex], [tex] P = 13(1 + w) [/tex]

C. [tex] 156 = 13 + w [/tex], [tex] P = 2(13 + w) [/tex]

D. [tex] 156 = 13 \times w [/tex], [tex] P = 2(13 + w) [/tex]

To solve the problem, we begin by using the information given:

1. Area of the rectangle: The formula for the area of a rectangle is:
[tex]\[
\text{Area} = \text{length} \times \text{width}
\][/tex]
We know the area is 156 square feet, and the length is 13 feet. So, we can set up the equation:
[tex]\[
156 = 13 \times w
\][/tex]
where [tex]\( w \)[/tex] represents the width.

2. Solving for the width:
To find the width, divide both sides of the equation by 13:
[tex]\[
w = \frac{156}{13} = 12
\][/tex]
So, the width of the room is 12 feet.

3. Finding the perimeter: The formula for the perimeter of a rectangle is:
[tex]\[
\text{Perimeter} = 2 \times (\text{length} + \text{width})
\][/tex]
Substitute the known values:
[tex]\[
\text{Perimeter} = 2 \times (13 + 12)
\][/tex]
[tex]\[
\text{Perimeter} = 2 \times 25 = 50
\][/tex]
Therefore, the perimeter of the room is 50 feet.

4. Choosing the correct system of equations: Based on the above calculations, the system of equations that matches our solution and incorporates both the area and the perimeter is:
[tex]\[
\text{D. } 156 = 13 \times w \text{ ; } P = 2(13 + w)
\][/tex]

Thus, the correct answer is D.