To solve the given problem, we'll follow the descriptions provided for each number and build up the equation step-by-step.
### Variables and Equations
- Let's denote the first number by \( x \).
- The second number is three more than twice the first number. Hence, the second number is \( 2x + 3 \).
- The third number is two more than the first number. Hence, the third number is \( x + 2 \).
### Setting up the Sum Equation
According to the problem, the sum of these three numbers is 45. Putting this all together, we get:
[tex]\[ x + (2x + 3) + (x + 2) = 45 \][/tex]
### Simplify the Equation
Now, we simplify the left-hand side of the equation step-by-step:
1. Combine all \( x \) terms: \( x + 2x + x \) which gives us \( 4x \).
2. Combine the constant terms: \( 3 + 2 \) which gives us \( 5 \).
Putting it all together, we get:
[tex]\[ 4x + 5 = 45 \][/tex]
### Solving for \( x \)
To solve for \( x \), we isolate the variable by following these steps:
1. Subtract 5 from both sides:
[tex]\[ 4x + 5 - 5 = 45 - 5 \][/tex]
[tex]\[ 4x = 40 \][/tex]
2. Divide both sides by 4:
[tex]\[ \frac{4x}{4} = \frac{40}{4} \][/tex]
[tex]\[ x = 10 \][/tex]
Thus, the equation that can be used to solve for the first number \( x \) is:
[tex]\[ x + (2x + 3) + (x + 2) = 45 \][/tex]
By substitution into your original multiple choice options, the correct equation is:
[tex]\[ x + 2x + 3 + x + 2 = 45 \][/tex]
### Matching with the Multiple Choice
While matching the options,
[tex]\[ x + 2 x + 3 + x + 2 = 45 \][/tex]
is the correct equation, which translates to:
[tex]\[ x + 2x + x + 3 + 2 = 45 \][/tex]
This format aligns with the given choices as:
[tex]\[ x + 2 x + 3 + x + 2 = 45 \][/tex]
Hence, the correct option from your provided list is:
[tex]\[ x + 2 x + 3 + x + 2 = 45 \][/tex]
