The sum of three numbers is 45. The second number is three more than twice the first number, \(x\). The third number is two more than the first number, \(x\). Which equation can be used to solve for the first number, \(x\)?

A. \(x + 5x + x + 2 = 45\)

B. \(x + x + 3 + x + 2 = 45\)

C. \(x + 3x + 2 + x + 2 = 45\)

D. [tex]\(x + 2x + 3 + x + 2 = 45\)[/tex]



Answer :

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]