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Which of the following equations has a solution different from the rest? (1 point)

Responses

3/4x+5=−1/4
Start Fraction 3 over 4 End Fraction x plus 5 equals negative Start Fraction 1 over 4 End Fraction

1/2x−3=1/2
Start Fraction 1 over 2 End Fraction x minus 3 equals Start Fraction 1 over 2 End Fraction

−0.35x−0.52=1.93
negative 0.35 x minus 0.52 equals 1.93

−1/7x−3/4=1/4
negative Start Fraction 1 over 7 End Fraction x minus Start Fraction 3 over 4 End Fraction equals Start Fraction 1 over 4 End Fraction

3 answers

To determine which equation has a solution different from the rest, we will solve each equation for \( x \).

  1. Equation 1: \[ \frac{3}{4}x + 5 = -\frac{1}{4} \] Subtract 5 from both sides: \[ \frac{3}{4}x = -\frac{1}{4} - 5 \] Convert 5 to a fraction with a denominator of 4: \[ 5 = \frac{20}{4} \quad \Rightarrow \quad \frac{3}{4}x = -\frac{1}{4} - \frac{20}{4} = -\frac{21}{4} \] Multiply both sides by \(\frac{4}{3}\): \[ x = -\frac{21}{4} \cdot \frac{4}{3} = -\frac{21}{3} = -7 \]

  2. Equation 2: \[ \frac{1}{2}x - 3 = \frac{1}{2} \] Add 3 to both sides: \[ \frac{1}{2}x = \frac{1}{2} + 3 \] Convert 3 to a fraction with a denominator of 2: \[ 3 = \frac{6}{2} \quad \Rightarrow \quad \frac{1}{2}x = \frac{1}{2} + \frac{6}{2} = \frac{7}{2} \] Multiply both sides by 2: \[ x = 7 \]

  3. Equation 3: \[ -0.35x - 0.52 = 1.93 \] Add 0.52 to both sides: \[ -0.35x = 1.93 + 0.52 = 2.45 \] Divide both sides by -0.35: \[ x = \frac{2.45}{-0.35} \approx -7 \]

  4. Equation 4: \[ -\frac{1}{7}x - \frac{3}{4} = \frac{1}{4} \] Add \(\frac{3}{4}\) to both sides: \[ -\frac{1}{7}x = \frac{1}{4} + \frac{3}{4} = 1 \] Multiply by -7: \[ x = -7 \]

Now, let's summarize the solutions:

  • Equation 1: \( x = -7 \)
  • Equation 2: \( x = 7 \)
  • Equation 3: \( x \approx -7 \)
  • Equation 4: \( x = -7 \)

The equation that has a solution different from the rest is Equation 2, which has a solution of \( x = 7 \). All the other equations have solutions either equal to or approximately equal to \(-7\).

which is correct

To find which equation has a solution different from the others, let's review the solutions one more time for clarity and correctness:

  1. Equation 1: \[ \frac{3}{4}x + 5 = -\frac{1}{4} \] Steps:

    • Subtract 5 from both sides: \[ \frac{3}{4}x = -\frac{1}{4} - 5 \]
    • Convert 5 to a fraction: \[ 5 = \frac{20}{4} \Rightarrow \frac{3}{4}x = -\frac{21}{4} \]
    • Multiply both sides by \(\frac{4}{3}\): \[ x = -\frac{21}{3} = -7 \]

    Solution: \( x = -7 \)

  2. Equation 2: \[ \frac{1}{2}x - 3 = \frac{1}{2} \] Steps:

    • Add 3 to both sides: \[ \frac{1}{2}x = \frac{1}{2} + 3 \]
    • Convert 3 to a fraction: \[ 3 = \frac{6}{2} \Rightarrow \frac{1}{2}x = \frac{7}{2} \]
    • Multiply both sides by 2: \[ x = 7 \]

    Solution: \( x = 7 \)

  3. Equation 3: \[ -0.35x - 0.52 = 1.93 \] Steps:

    • Add 0.52 to both sides: \[ -0.35x = 1.93 + 0.52 = 2.45 \]
    • Divide by -0.35: \[ x = \frac{2.45}{-0.35} \approx -7 \]

    Solution: \( x \approx -7 \)

  4. Equation 4: \[ -\frac{1}{7}x - \frac{3}{4} = \frac{1}{4} \] Steps:

    • Add \(\frac{3}{4}\) to both sides: \[ -\frac{1}{7}x = \frac{1}{4} + \frac{3}{4} = 1 \]
    • Multiply by -7: \[ x = -7 \]

    Solution: \( x = -7 \)

Summary of Solutions

  • Equation 1: \( x = -7 \)
  • Equation 2: \( x = 7 \)
  • Equation 3: \( x \approx -7 \)
  • Equation 4: \( x = -7 \)

Conclusion

The equation with a solution different from the rest is Equation 2: \(\frac{1}{2}x - 3 = \frac{1}{2}\), which has the solution \( x = 7 \). The other equations all yield solutions approximately equal to \(-7\).