To determine which of the given equations has a solution different from the rest, we need to solve each equation for \( x \) and compare the solutions.
-
Equation 1: \( 1.2x - 3 = \frac{1}{2} \)
\[ 1.2x = \frac{1}{2} + 3 \] \[ 1.2x = \frac{1}{2} + \frac{6}{2} = \frac{7}{2} \] \[ x = \frac{7/2}{1.2} = \frac{7}{2} \cdot \frac{10}{12} = \frac{7 \cdot 5}{12} = \frac{35}{12} \approx 2.917 \]
-
Equation 2: \( 3.4x + 5 = -\frac{1}{4} \)
\[ 3.4x = -\frac{1}{4} - 5 \] \[ 3.4x = -\frac{1}{4} - \frac{20}{4} = -\frac{21}{4} \] \[ x = -\frac{21/4}{3.4} = -\frac{21/4}{17/5} = -\frac{21 \cdot 5}{4 \cdot 17} = -\frac{105}{68} \]
-
Equation 3: \( -0.35x - 0.52 = 1.93 \)
\[ -0.35x = 1.93 + 0.52 \] \[ -0.35x = 2.45 \] \[ x = \frac{2.45}{-0.35} = -7 \]
-
Equation 4: \( -\frac{1}{7}x - \frac{3}{4} = \frac{1}{4} \)
\[ -\frac{1}{7}x = \frac{1}{4} + \frac{3}{4} \] \[ -\frac{1}{7}x = 1 \] \[ x = -7 \]
Now we have the solutions for the equations:
- \( x \approx 2.917 \)
- \( x = -\frac{105}{68} \) (approximately -1.544)
- \( x = -7 \)
- \( x = -7 \)
The first equation has a solution of approximately \( 2.917 \), which is different from the other three equations, which all have a solution of \( -7 \) (or closely related to it).
Therefore, the equation with a unique solution is:
1. \( 1.2x - 3 = \frac{1}{2} \)
1 answer