Felix wrote several equations and determined that only one of the equations has no solution. Which of these equations has no solution?

Question

3 (x minus 2) + x = 4 x + 6
3 (x minus 2) + x = 4 x minus 6
3 (x minus 2) + x = 2 x minus 6
3 (x minus 2) + x = 3 x minus 3

Answer 1

To determine which equation has no solution, we'll simplify each equation and see if any leads to a contradiction (where both sides can't be equal).

Let's analyze each equation one by one.

1. Equation 1: \( 3(x - 2) + x = 4x + 6 \)
   First, distribute the left side: \[ 3x - 6 + x = 4x + 6 \] Combine like terms: \[ 4x - 6 = 4x + 6 \] Subtract \(4x\) from both sides: \[ -6 = 6 \] This is a contradiction, meaning this equation has no solution.

2. Equation 2: \( 3(x - 2) + x = 4x - 6 \)
   Again, distribute: \[ 3x - 6 + x = 4x - 6 \] Combine like terms: \[ 4x - 6 = 4x - 6 \] This simplifies to \(0 = 0\), which is always true. Thus, this equation has infinite solutions.

3. Equation 3: \( 3(x - 2) + x = 2x - 6 \)
   Distribute: \[ 3x - 6 + x = 2x - 6 \] Combine like terms: \[ 4x - 6 = 2x - 6 \] Subtract \(2x\) from both sides: \[ 2x - 6 = -6 \] Adding 6 to both sides: \[ 2x = 0 \Rightarrow x = 0 \] There is one solution: \(x = 0\).

4. Equation 4: \( 3(x - 2) + x = 3x - 3 \)
   Distribute: \[ 3x - 6 + x = 3x - 3 \] Combine like terms: \[ 4x - 6 = 3x - 3 \] Subtract \(3x\) from both sides: \[ x - 6 = -3 \] Adding 6 to both sides: \[ x = 3 \] There is one solution: \(x = 3\).

Conclusion: The equation with no solution is \( 3(x - 2) + x = 4x + 6 \).