Answer :
Let's start by examining the problem step by step.
### Step 1: Given Data
We know that the perimeter of the first rectangle is represented by the expression:
[tex]\[6x + 8\][/tex]
### Step 2: Expressions for the Second Rectangle
We are given several possible expressions for the perimeter of the second rectangle:
1. [tex]\(18x + 4y + 7\)[/tex]
2. [tex]\(18x + 2y + 6\)[/tex]
3. [tex]\(22xy + 6\)[/tex]
4. [tex]\(24x + 2y + 10\)[/tex]
### Step 3: Selecting Valid Perimeter Expressions
To determine which expression properly represents the perimeter of a rectangle, recall that the perimeter formula [tex]\(P\)[/tex] for a rectangle with length [tex]\(l\)[/tex] and width [tex]\(w\)[/tex] is given by:
[tex]\[P = 2l + 2w\][/tex]
Therefore, the valid expressions should be linear combinations of the dimensions. Expression 3, [tex]\(22xy + 6\)[/tex], is not linear due to the multiplication of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Let's consider the valid remaining expressions:
1. [tex]\(18x + 4y + 7\)[/tex]
2. [tex]\(18x + 2y + 6\)[/tex]
4. [tex]\(24x + 2y + 10\)[/tex]
### Step 4: Calculating Combined Perimeters
We now need to find the combined perimeter of the first and second rectangles using the selected expressions.
1. First Combination:
- Perimeter of first rectangle: [tex]\(6x + 8\)[/tex]
- Perimeter of second rectangle: [tex]\(18x + 4y + 7\)[/tex]
- Combined perimeter: [tex]\( (6x + 8) + (18x + 4y + 7)\)[/tex]
- Simplifying this:
[tex]\[ 6x + 18x + 4y + 8 + 7 = 24x + 4y + 15 \][/tex]
- Result: [tex]\((24x, 4y, 15)\)[/tex]
2. Second Combination:
- Perimeter of first rectangle: [tex]\(6x + 8\)[/tex]
- Perimeter of second rectangle: [tex]\(18x + 2y + 6\)[/tex]
- Combined perimeter: [tex]\( (6x + 8) + (18x + 2y + 6)\)[/tex]
- Simplifying this:
[tex]\[ 6x + 18x + 2y + 8 + 6 = 24x + 2y + 14 \][/tex]
- Result: [tex]\((24x, 2y, 14)\)[/tex]
3. Third Combination:
- Perimeter of first rectangle: [tex]\(6x + 8\)[/tex]
- Perimeter of second rectangle: [tex]\(24x + 2y + 10\)[/tex]
- Combined perimeter: [tex]\( (6x + 8) + (24x + 2y + 10)\)[/tex]
- Simplifying this:
[tex]\[ 6x + 24x + 2y + 8 + 10 = 30x + 2y + 18 \][/tex]
- This is less logical given the actual combined result found, specifically looking at the correct combined perimeter expressed below.
Hence, while the specific values for processes 1 and 2 above are kept closer to the combined perimeter logics.
### Step 5: Final Combined Expression
The total combined perimeter can then be summarized effectively ensuring it's a dual combination:
[tex]\[ \text{Combined Perimeter: }\quad (48, 2, 18) \][/tex]
### Conclusion
Hence our chosen perimeter expressions and their evaluated combined perimeter should match each section respectively:
1. [tex]\( (24, 4, 15) \)[/tex]
2. [tex]\( (24, 2, 14) \)[/tex]
3. Ensuring robust combined total [tex]\( (48, 2, 18) \)[/tex].
Thus, comparing, resulting gives these useful evaluating points for combined perimeter values further ensuring the proper combination.
### Step 1: Given Data
We know that the perimeter of the first rectangle is represented by the expression:
[tex]\[6x + 8\][/tex]
### Step 2: Expressions for the Second Rectangle
We are given several possible expressions for the perimeter of the second rectangle:
1. [tex]\(18x + 4y + 7\)[/tex]
2. [tex]\(18x + 2y + 6\)[/tex]
3. [tex]\(22xy + 6\)[/tex]
4. [tex]\(24x + 2y + 10\)[/tex]
### Step 3: Selecting Valid Perimeter Expressions
To determine which expression properly represents the perimeter of a rectangle, recall that the perimeter formula [tex]\(P\)[/tex] for a rectangle with length [tex]\(l\)[/tex] and width [tex]\(w\)[/tex] is given by:
[tex]\[P = 2l + 2w\][/tex]
Therefore, the valid expressions should be linear combinations of the dimensions. Expression 3, [tex]\(22xy + 6\)[/tex], is not linear due to the multiplication of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Let's consider the valid remaining expressions:
1. [tex]\(18x + 4y + 7\)[/tex]
2. [tex]\(18x + 2y + 6\)[/tex]
4. [tex]\(24x + 2y + 10\)[/tex]
### Step 4: Calculating Combined Perimeters
We now need to find the combined perimeter of the first and second rectangles using the selected expressions.
1. First Combination:
- Perimeter of first rectangle: [tex]\(6x + 8\)[/tex]
- Perimeter of second rectangle: [tex]\(18x + 4y + 7\)[/tex]
- Combined perimeter: [tex]\( (6x + 8) + (18x + 4y + 7)\)[/tex]
- Simplifying this:
[tex]\[ 6x + 18x + 4y + 8 + 7 = 24x + 4y + 15 \][/tex]
- Result: [tex]\((24x, 4y, 15)\)[/tex]
2. Second Combination:
- Perimeter of first rectangle: [tex]\(6x + 8\)[/tex]
- Perimeter of second rectangle: [tex]\(18x + 2y + 6\)[/tex]
- Combined perimeter: [tex]\( (6x + 8) + (18x + 2y + 6)\)[/tex]
- Simplifying this:
[tex]\[ 6x + 18x + 2y + 8 + 6 = 24x + 2y + 14 \][/tex]
- Result: [tex]\((24x, 2y, 14)\)[/tex]
3. Third Combination:
- Perimeter of first rectangle: [tex]\(6x + 8\)[/tex]
- Perimeter of second rectangle: [tex]\(24x + 2y + 10\)[/tex]
- Combined perimeter: [tex]\( (6x + 8) + (24x + 2y + 10)\)[/tex]
- Simplifying this:
[tex]\[ 6x + 24x + 2y + 8 + 10 = 30x + 2y + 18 \][/tex]
- This is less logical given the actual combined result found, specifically looking at the correct combined perimeter expressed below.
Hence, while the specific values for processes 1 and 2 above are kept closer to the combined perimeter logics.
### Step 5: Final Combined Expression
The total combined perimeter can then be summarized effectively ensuring it's a dual combination:
[tex]\[ \text{Combined Perimeter: }\quad (48, 2, 18) \][/tex]
### Conclusion
Hence our chosen perimeter expressions and their evaluated combined perimeter should match each section respectively:
1. [tex]\( (24, 4, 15) \)[/tex]
2. [tex]\( (24, 2, 14) \)[/tex]
3. Ensuring robust combined total [tex]\( (48, 2, 18) \)[/tex].
Thus, comparing, resulting gives these useful evaluating points for combined perimeter values further ensuring the proper combination.
