To determine which of the equations has infinitely many solutions, we need to look for an identity, which is an equation that is true for all values of \( x \).
Let's analyze each equation:
-
Equation 1: \( 3 - 3x = -3(x - 1) \)
Expanding the right side: \[ 3 - 3x = -3x + 3 \] Rearranging: \[ 3 - 3x + 3x = 3 \quad \text{(true for any } x\text{)} \] Thus, this equation is an identity and has infinitely many solutions.
-
Equation 2: \( 3x = 2x + 1 \)
Rearranging gives: \[ 3x - 2x = 1 \implies x = 1 \] This equation has a single solution.
-
Equation 3: \( 3x + 5 = 3x - 5 \)
Rearranging gives: \[ 5 = -5 \quad \text{(not true)} \] This equation has no solutions.
-
Equation 4: \( 4x + 1 = 4(x + 1) \)
Expanding the right side: \[ 4x + 1 = 4x + 4 \] Rearranging gives: \[ 1 = 4 \quad \text{(not true)} \] This equation has no solutions.
Based on this analysis, the equation with infinitely many solutions is:
Answer: \( 3 - 3x = -3(x - 1) \)