Which of the following equations has infinitely many solutions?*1 point2x + 3 = 5 + 2x2x + 3 = 5 + 3x3x - 5 = -5 + 2x3x - 5 = -5 + 3x

1. Break Down the Problem

We need to analyze each equation to determine which one has infinitely many solutions. An equation has infinitely many solutions if both sides are identical, leading to a true statement, regardless of the value of the variable.

2. Relevant Concepts

To check for infinitely many solutions, we will simplify each equation to see if they can be transformed into a form where both sides are the same.

3. Analysis and Detail

1. Equation: $2x + 3 = 5 + 2x$
   Rearranging gives:
   $2x - 2x = 5 - 3$
   $0 = 2$ (False). No solutions.

2. Equation: $2x + 3 = 5 + 3x$
   Rearranging gives:
   $2x - 3x = 5 - 3$
   $-x = 2$ (Single solution: $x = -2$).

3. Equation: $3x - 5 = -5 + 2x$
   Rearranging gives:
   $3x - 2x = -5 + 5$
   $x = 0$ (Single solution: $x = 0$).

4. Equation: $3x - 5 = -5 + 3x$
   Rearranging gives:
   $3x - 3x = -5 + 5$
   $0 = 0$ (True for all $x$). Infinitely many solutions.

4. Verify and Summarize

Out of the four equations, the fourth equation $3x - 5 = -5 + 3x$ simplifies to $0 = 0$, indicating that it has infinitely many solutions.

Final Answer

The equation with infinitely many solutions is $3x - 5 = -5 + 3x$.