Solve for X in the Fraction Equation: Isolate the Variable Step-by-Step

To solve this linear equation, we begin by simplifying it:

Step 1: Distribute the fraction on the left side of the equation.

• We have $-\frac{1}{5}(x + \frac{1}{4})$. Distribute the $-\frac{1}{5}$:
  $-\frac{1}{5} \cdot x - \frac{1}{5} \cdot \frac{1}{4} = -\frac{1}{5}x - \frac{1}{20}$ So, the equation becomes:
  $-\frac{1}{5}x - \frac{1}{20} = \frac{7}{10} + \frac{3}{5}x - \frac{2}{5}$

Step 2: Simplify the right side.

• First, combine the constant terms $\frac{7}{10}$ and $-\frac{2}{5}$ on the right side:
  Convert $-\frac{2}{5}$ to tenths to combine: $-\frac{2}{5} = -\frac{4}{10}$.
  Now, $\frac{7}{10} - \frac{4}{10} = \frac{3}{10}$.
  The right side simplifies to:
  $\frac{3}{10} + \frac{3}{5}x$

Step 3: Move all terms involving $x$ to one side and constants to the other.

• Add $\frac{1}{5}x$ to both sides:
  $-\frac{1}{20} = \frac{3}{10} + \frac{3}{5}x + \frac{1}{5}x$
• Combine like terms involving $x$ on the right side:
  Convert $\frac{1}{5}x$ to a common denominator with $\frac{3}{5}x$ which is $\frac{3}{5}x = \frac{6}{10}x$ and $\frac{1}{5}x = \frac{2}{10}x$, giving us:
  $\frac{8}{10}x$, thus yielding
  $\frac{3}{10} + \frac{8}{10}x$
• To isolate $x$, subtract $\frac{3}{10}$ from both sides:
  $-\frac{1}{20} - \frac{3}{10} = \frac{8}{10}x$

Step 4: Solve the resulting equation for $x$.

• Calculate $-\frac{1}{20} - \frac{3}{10}$:
  Convert $-\frac{3}{10}$ to a common denominator with $-\frac{1}{20}$:
  $-\frac{3}{10} = -\frac{6}{20}$.
  So, $-\frac{1}{20} - \frac{6}{20} = -\frac{7}{20}$:
  The equation becomes:
  $-\frac{7}{20} = \frac{8}{10}x$
• Divide both sides by $\frac{8}{10}$ to solve for $x$:
  $x = \left(-\frac{7}{20}\right) \div \left(\frac{8}{10}\right)$
• When dividing fractions, invert the divisor and multiply:
  $x = -\frac{7}{20} \times \frac{10}{8}$
• Simplify:
  $x = -\frac{7 \times 10}{20 \times 8} = -\frac{70}{160}$
• Simplify further by dividing both numerator and denominator by the greatest common divisor (which is 10):
  $x = -\frac{7}{16}$

Therefore, the solution to the equation is $x = -\frac{7}{16}$.

The correct choice is:

$-\frac{7}{16}$