To solve the equation
\[ \frac{2}{3} + \frac{1}{5}x - \frac{1}{4} = \frac{4}{5}x, \]
let's first rearrange the equation to isolate all terms involving \(x\) on one side and constant terms on the other side.
- Start by simplifying the equation:
\[ \frac{1}{5}x - \frac{4}{5}x = -\frac{2}{3} + \frac{1}{4}. \]
- Combine the \(x\) terms:
\[ \frac{1 - 4}{5}x = -\frac{2}{3} + \frac{1}{4}. \]
This simplifies to:
\[ -\frac{3}{5}x = -\frac{2}{3} + \frac{1}{4}. \]
- Now we need a common denominator to combine the constants on the right side. The least common multiple of 3 and 4 is 12. Rewrite \(-\frac{2}{3}\) and \(\frac{1}{4}\):
\[ -\frac{2}{3} = -\frac{8}{12}, \] \[ \frac{1}{4} = \frac{3}{12}. \]
So we have:
\[ -\frac{3}{5}x = -\frac{8}{12} + \frac{3}{12}. \]
Now combine the fractions on the right:
\[ -\frac{3}{5}x = -\frac{8 - 3}{12} = -\frac{5}{12}. \]
- Now we can multiply both sides by -1:
\[ \frac{3}{5}x = \frac{5}{12}. \]
- To isolate \(x\), multiply both sides by the reciprocal of \(\frac{3}{5}\), which is \(\frac{5}{3}\):
\[ x = \frac{5}{12} \cdot \frac{5}{3} = \frac{25}{36}. \]
Thus, the solution to the equation is
\[ \boxed{\frac{25}{36}}. \]
1 answer