To determine which equation has infinitely many solutions, we need to simplify each given equation and check if they are true for all values of \( x \).
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Equation 1: \( 3x + 5 = 3x - 5 \) \[ 3x + 5 - 3x = -5 \implies 5 = -5 \quad \text{(not true)} \] This equation has no solutions.
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Equation 2: \( 3 - 3x = -3(x - 1) \) \[ 3 - 3x = -3x + 3 \] By adding \( 3x \) to both sides, we get: \[ 3 = 3 \quad \text{(true for any value of \( x \))} \] This equation has infinitely many solutions.
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Equation 3: \( 3x = 2x + 1 \) \[ 3x - 2x = 1 \implies x = 1 \quad \text{(one solution)} \]
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Equation 4: \( 4x + 1 = 4(x + 1) \) Expanding the right side: \[ 4x + 1 = 4x + 4 \] Subtracting \( 4x \) from both sides gives: \[ 1 = 4 \quad \text{(not true)} \] This equation has no solutions.
The only equation with infinitely many solutions is Equation 2: \( 3 - 3x = -3(x - 1) \).