To determine which equation has a different solution than the others, we will solve each equation for \( x \).
Equation A:
\[
-\frac{1}{7}x - \frac{3}{4} = \frac{1}{4}
\]
Adding \(\frac{3}{4}\) to both sides:
\[
-\frac{1}{7}x = \frac{1}{4} + \frac{3}{4} = 1
\]
Multiplying both sides by \(-7\):
\[
x = -7
\]
Equation B: \[ -0.35x - 0.52 = 1.93 \] Adding \(0.52\) to both sides: \[ -0.35x = 1.93 + 0.52 = 2.45 \] Dividing by \(-0.35\): \[ x = \frac{2.45}{-0.35} = -7 \]
Equation C: \[ \frac{1}{2}x - 3 = \frac{1}{2} \] Adding \(3\) to both sides: \[ \frac{1}{2}x = \frac{1}{2} + 3 = \frac{1}{2} + \frac{6}{2} = \frac{7}{2} \] Multiplying both sides by \(2\): \[ x = 7 \]
Equation D: \[ \frac{3}{4}x + 5 = -\frac{1}{4} \] Subtracting \(5\) from both sides: \[ \frac{3}{4}x = -\frac{1}{4} - 5 = -\frac{1}{4} - \frac{20}{4} = -\frac{21}{4} \] Multiplying both sides by \(\frac{4}{3}\): \[ x = -\frac{21}{4} \cdot \frac{4}{3} = -\frac{21}{3} = -7 \]
Now, let's summarize the solutions:
- A: \(x = -7\)
- B: \(x = -7\)
- C: \(x = 7\)
- D: \(x = -7\)
The solution for equation C is different from the rest, as it results in \( 7 \) instead of \( -7 \).
Therefore, the equation that has a different solution is C: \(\frac{1}{2}x - 3 = \frac{1}{2}\).