Question

Felix wrote several equations and determined that only one of the equations has no solution. Which of these equations has no solution?
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 \( 3(x - 2) + x = 4x + 6 \).