Answer :
To determine which equation models the problem correctly, let's compare each equation to see if it satisfies the condition that the sum of the three sides equals 23.
### Given Options:
1. [tex]\(x + x + (5 - 2x) = 23\)[/tex]
2. [tex]\(x + x + (2x - 5) = 23\)[/tex]
3. [tex]\(x + x + (2x + 5) = 23\)[/tex]
4. [tex]\(x + (2x - 5) + (2x - 5) = 23\)[/tex]
We'll analyze these equations one by one.
### Option 1: [tex]\(x + x + (5 - 2x) = 23\)[/tex]
Simplifying this equation:
[tex]\[x + x + (5 - 2x) = 23\][/tex]
[tex]\[2x + 5 - 2x = 23\][/tex]
[tex]\[5 = 23\][/tex]
This is not correct, since 5 does not equal 23. Therefore, this equation does not model the problem.
### Option 2: [tex]\(x + x + (2x - 5) = 23\)[/tex]
Simplifying this equation:
[tex]\[ x + x + (2x - 5) = 23 \][/tex]
[tex]\[ 2x + 2x - 5 = 23 \][/tex]
[tex]\[ 4x - 5 = 23 \][/tex]
This equation is valid and simplifies to [tex]\(4x - 5 = 23\)[/tex].
### Option 3: [tex]\(x + x + (2x + 5) = 23\)[/tex]
Simplifying this equation:
[tex]\[ x + x + (2x + 5) = 23 \][/tex]
[tex]\[ 2x + 2x + 5 = 23 \][/tex]
[tex]\[ 4x + 5 = 23 \][/tex]
This equation is valid and simplifies to [tex]\(4x + 5 = 23\)[/tex].
### Option 4: [tex]\(x + (2x - 5) + (2x - 5) = 23\)[/tex]
Simplifying this equation:
[tex]\[ x + (2x - 5) + (2x - 5) = 23 \][/tex]
[tex]\[ x + 2x - 5 + 2x - 5 = 23 \][/tex]
[tex]\[ x + 4x - 10 = 23 \][/tex]
[tex]\[ 5x - 10 = 23 \][/tex]
This equation is valid and simplifies to [tex]\(5x - 10 = 23\)[/tex].
### Summary
Based on the analysis:
- Option 1 simplifies to [tex]\(5 = 23\)[/tex], which is incorrect.
- Option 2 simplifies to [tex]\(4x - 5 = 23\)[/tex].
- Option 3 simplifies to [tex]\(4x + 5 = 23\)[/tex].
- Option 4 simplifies to [tex]\(5x - 10 = 23\)[/tex].
Therefore, the correct equations that can model the problem are:
[tex]\(x + x + (2x - 5) = 23\)[/tex],
[tex]\(x + x + (2x + 5) = 23\)[/tex], and
[tex]\(x + (2x - 5) + (2x - 5) = 23\)[/tex].
### Given Options:
1. [tex]\(x + x + (5 - 2x) = 23\)[/tex]
2. [tex]\(x + x + (2x - 5) = 23\)[/tex]
3. [tex]\(x + x + (2x + 5) = 23\)[/tex]
4. [tex]\(x + (2x - 5) + (2x - 5) = 23\)[/tex]
We'll analyze these equations one by one.
### Option 1: [tex]\(x + x + (5 - 2x) = 23\)[/tex]
Simplifying this equation:
[tex]\[x + x + (5 - 2x) = 23\][/tex]
[tex]\[2x + 5 - 2x = 23\][/tex]
[tex]\[5 = 23\][/tex]
This is not correct, since 5 does not equal 23. Therefore, this equation does not model the problem.
### Option 2: [tex]\(x + x + (2x - 5) = 23\)[/tex]
Simplifying this equation:
[tex]\[ x + x + (2x - 5) = 23 \][/tex]
[tex]\[ 2x + 2x - 5 = 23 \][/tex]
[tex]\[ 4x - 5 = 23 \][/tex]
This equation is valid and simplifies to [tex]\(4x - 5 = 23\)[/tex].
### Option 3: [tex]\(x + x + (2x + 5) = 23\)[/tex]
Simplifying this equation:
[tex]\[ x + x + (2x + 5) = 23 \][/tex]
[tex]\[ 2x + 2x + 5 = 23 \][/tex]
[tex]\[ 4x + 5 = 23 \][/tex]
This equation is valid and simplifies to [tex]\(4x + 5 = 23\)[/tex].
### Option 4: [tex]\(x + (2x - 5) + (2x - 5) = 23\)[/tex]
Simplifying this equation:
[tex]\[ x + (2x - 5) + (2x - 5) = 23 \][/tex]
[tex]\[ x + 2x - 5 + 2x - 5 = 23 \][/tex]
[tex]\[ x + 4x - 10 = 23 \][/tex]
[tex]\[ 5x - 10 = 23 \][/tex]
This equation is valid and simplifies to [tex]\(5x - 10 = 23\)[/tex].
### Summary
Based on the analysis:
- Option 1 simplifies to [tex]\(5 = 23\)[/tex], which is incorrect.
- Option 2 simplifies to [tex]\(4x - 5 = 23\)[/tex].
- Option 3 simplifies to [tex]\(4x + 5 = 23\)[/tex].
- Option 4 simplifies to [tex]\(5x - 10 = 23\)[/tex].
Therefore, the correct equations that can model the problem are:
[tex]\(x + x + (2x - 5) = 23\)[/tex],
[tex]\(x + x + (2x + 5) = 23\)[/tex], and
[tex]\(x + (2x - 5) + (2x - 5) = 23\)[/tex].
