Solve the Fraction Equation: Determining X in 36 - 18/20x + 5/10x = 23 + 15 - 1/5x

To solve the equation $36 - \frac{18}{20}x + \frac{5}{10}x = 23 + 15 - \frac{1}{5}x$, we will follow these steps:

• Simplify the fractional coefficients for $x$.
• Combine like terms on both sides of the equation.
• Isolate the terms involving $x$ on one side and constants on the other side.
• Solve for $x$.

Let's proceed with these steps:

Step 1: Simplify the fractional coefficients.
- $\frac{18}{20}$ simplifies to $\frac{9}{10}$.
- $\frac{5}{10}$ simplifies to $\frac{1}{2}$.
- $\frac{1}{5}$ is already simplified.

Step 2: Rewrite the original equation using these simplifications:
$36 - \frac{9}{10}x + \frac{1}{2}x = 23 + 15 - \frac{1}{5}x$.

Step 3: Combine the constants and terms involving $x$.
On the left side, combine $-\frac{9}{10}x$ and $+\frac{1}{2}x$:
- Convert $\frac{1}{2}$ to a denominator of 10: $\frac{1}{2} = \frac{5}{10}$.
- Thus, $-\frac{9}{10}x + \frac{5}{10}x = -\frac{4}{10}x = -\frac{2}{5}x$.

The equation becomes:
$36 - \frac{2}{5}x = 38 - \frac{1}{5}x$.

Step 4: Move all terms involving $x$ to one side of the equation:
Add $\frac{1}{5}x$ to both sides:
$36 - \frac{2}{5}x + \frac{1}{5}x = 38$.
This simplifies to:
$36 - \frac{1}{5}x = 38$.

Step 5: Isolate $-\frac{1}{5}x$:
Subtract 36 from both sides:
$-\frac{1}{5}x = 2$.

Step 6: Solve for $x$ by multiplying both sides by $-5$:
$x = -10$.

Therefore, the solution to the equation is $x = -10$.