Solve the Fraction Equation with Negative Fractions: Finding X in -1/5(x-1/3)+1/15=-3/5x+1/10

To solve this problem, we'll follow these steps:

• Step 1: Distribute $-\frac{1}{5}$ across $(x-\frac{1}{3})$.
• Step 2: Clear fractions by multiplying through by the least common multiple (LCM) of the denominators.
• Step 3: Simplify and combine like terms to isolate $x$.

Let's work through each step:

Step 1: Distribute $-\frac{1}{5}$ across $(x-\frac{1}{3})$.
$-\frac{1}{5}(x-\frac{1}{3}) = -\frac{1}{5}x + \frac{1}{15}$.

The equation now is:
$-\frac{1}{5}x + \frac{1}{15} + \frac{1}{15} = -\frac{3}{5}x + \frac{1}{10}$.

Simplify the left side:
$-\frac{1}{5}x + \frac{2}{15} = -\frac{3}{5}x + \frac{1}{10}$.

Step 2: Multiply through by 30, which is the LCM of 5, 15, and 10, to clear fractions.
$30(-\frac{1}{5}x) + 30(\frac{2}{15}) = 30(-\frac{3}{5}x) + 30(\frac{1}{10})$.

This gives us:
$-6x + 4 = -18x + 3$.

Step 3: Solve for $x$.
Add $18x$ to both sides to get:
$12x + 4 = 3$.

Subtract 4 from both sides:
$12x = -1$.

Divide both sides by 12:
$x = -\frac{1}{12}$.

Therefore, the solution to the problem is $x = -\frac{1}{12}$.