Solving for X with Fractional Coefficients: Equation Breakdown

To solve the equation $\frac{1}{5}x - \frac{2}{3} + \frac{1}{4}x = \frac{3}{4}x - \frac{3}{5} + \frac{1}{5}$, we will follow these steps:

• Step 1: Combine like terms on both sides.

• Step 2: Move all $x$-related terms to one side and constant terms to the other side.

• Step 3: Solve for $x$.

Let's apply these steps:

Step 1: Combine Like Terms
On the left side: $\frac{1}{5}x + \frac{1}{4}x = \frac{4}{20}x + \frac{5}{20}x = \frac{9}{20}x$
The left side becomes: $\frac{9}{20}x - \frac{2}{3}$.
On the right side: $\frac{3}{4}x = \frac{15}{20}x$, leaving $\frac{15}{20}x - \frac{3}{5} + \frac{1}{5}$.
Combine constants: $-\frac{3}{5} + \frac{1}{5} = -\frac{2}{5}$, so the right becomes: $\frac{15}{20}x - \frac{2}{5}$.

Step 2: Isolate $x$ Terms
Rearrange the equation: $\frac{9}{20}x - \frac{2}{3} = \frac{15}{20}x - \frac{2}{5}$.
Add $\frac{2}{3}$ to both sides:
$\frac{9}{20}x = \frac{15}{20}x - \frac{2}{5} + \frac{2}{3}$.

Convert $-\frac{2}{5}$ and $\frac{2}{3}$ to common denominators:
$-\frac{2}{5} = -\frac{24}{60}$ and $\frac{2}{3} = \frac{40}{60}$.
So, $-\frac{2}{5} + \frac{2}{3} = \frac{16}{60} = \frac{4}{15}$.

Thus, we have:
$\frac{9}{20}x = \frac{15}{20}x + \frac{4}{15}$.

Subtract $\frac{15}{20}x$ from both sides:
$\frac{9}{20}x - \frac{15}{20}x = \frac{4}{15}$.
This simplifies to $-\frac{6}{20}x = \frac{4}{15}$, or $-\frac{3}{10}x = \frac{4}{15}$.

Step 3: Solve for $x$
Multiply both sides by $-10/3$:
$x = \frac{4}{15} \times -\frac{10}{3}$.
This results in $x = -\frac{40}{45}$, which simplifies to $-\frac{8}{9}$.

Therefore, the solution to the problem is $x = -\frac{8}{9}$.