Certainly! Let's work through this step-by-step.
### Step 1: Draw and Label a Rectangle
First, let's draw a rectangle.
```
x (Length)
+-----------+
| |
| | Width: (x - 7)
| |
+-----------+
```
Here, the length of the rectangle is labeled as [tex]\( x \)[/tex] feet and the width is [tex]\( x - 7 \)[/tex] feet.
### Step 2: Write Three Different Expressions for the Perimeter
The perimeter [tex]\( P \)[/tex] of a rectangle can be found using the formula:
[tex]\[ P = 2 \times (\text{Length} + \text{Width}) \][/tex]
#### Expression 1: Using the Perimeter Formula Directly
[tex]\[ P_1 = 2 \times (x + (x - 7)) \][/tex]
[tex]\[ P_1 = 2 \times (2x - 7) \][/tex]
[tex]\[ P_1 = 4x - 14 \][/tex]
#### Expression 2: Summing Up all Sides Separately
The perimeter is also the sum of all the sides.
[tex]\[ P_2 = x + (x - 7) + x + (x - 7) \][/tex]
[tex]\[ P_2 = 4x - 14 \][/tex]
#### Expression 3: Using an Alternative Form of the Perimeter Formula
Another way to think about the perimeter is the sum of two lengths and two widths.
[tex]\[ P_3 = 2x + 2(x - 7) \][/tex]
[tex]\[ P_3 = 2x + 2x - 14 \][/tex]
[tex]\[ P_3 = 4x - 14 \][/tex]
### Final Simplified Expressions
So, the three different expressions for the perimeter of the rectangle are:
[tex]\[ P_1 = 4x - 14 \][/tex]
[tex]\[ P_2 = 4x - 14 \][/tex]
[tex]\[ P_3 = 4x - 14 \][/tex]
Thus, regardless of the method used, the perimeter of the rectangle simplifies to [tex]\( 4x - 14 \)[/tex] feet.
