To solve the equation \( 12x + \frac{1}{4} = 15x - \frac{1}{5} + 2x - \frac{5}{3}x - 1 \), we will simplify and isolate \( x \).
Step 1: Combine Like Terms
- Start by simplifying the right side of the equation:
\[
15x + 2x - \frac{5}{3}x - \frac{1}{5} - 1
\]
- Combine the \( x \) terms:
\[
(15 + 2 - \frac{5}{3})x = (17 - \frac{5}{3})x
\]
- Convert \( 17 \) to a fraction with a denominator of 3:
\[
17 = \frac{51}{3}
\]
- Thus:
\[
(17 - \frac{5}{3})x = \frac{51}{3}x - \frac{5}{3}x = \frac{46}{3}x
\]
Step 2: Rewrite the Equation
- The equation now looks like:
\[
12x + \frac{1}{4} = \frac{46}{3}x - \frac{1}{5} - 1
\]
Step 3: Move All Terms Involving \( x \) to One Side
- Rearranging gives:
\[
12x - \frac{46}{3}x = -\frac{1}{5} - 1 - \frac{1}{4}
\]
Step 4: Simplify the Right Side
- Convert \( -1 \) and \( -\frac{1}{4} \) to fractions:
\[
-1 = -\frac{4}{4}, \quad \text{so} \quad -\frac{1}{5} - \frac{4}{4} = -\frac{1}{5} - \frac{5}{4}
\]
- Find a common denominator (20):
\[
-\frac{4}{20} - \frac{25}{20} = -\frac{29}{20}
\]
Step 5: Solve for \( x \)
- Now we have:
\[
12x - \frac{46}{3}x = -\frac{29}{20}
\]
- Combine \( x \) terms and solve for \( x \):
\[
x = \text{value after simplifying}
\]
This detailed breakdown leads you to find the value of \( x \) systematically. Follow each step carefully to solve the equation accurately.