Understanding the Equation
To solve the equation \(12x + \frac{1}{4} = 15x - \frac{1}{5} + 2x - \frac{5}{3}x - 1\), let's simplify each side step by step.
Step 1: Simplifying Both Sides
- Start by combining like terms on the right side.
- Rewrite the equation:
\(12x + \frac{1}{4} = (15x + 2x - \frac{5}{3}x) - \frac{1}{5} - 1\)
- Combine the \(x\) terms:
\(15x + 2x - \frac{5}{3}x = 17x - \frac{5}{3}x\)
Convert \(17x\) to have a denominator of 3:
\((\frac{51}{3}x - \frac{5}{3}x) = \frac{46}{3}x\)
- Thus, the right side becomes:
\(\frac{46}{3}x - \frac{1}{5} - 1\)
Step 2: Consolidate Constants
- Convert -1 to a fraction:
\(-1 = -\frac{5}{5}\)
- Combine constants:
\(-\frac{1}{5} - \frac{5}{5} = -\frac{6}{5}\)
- The equation now looks like:
\(12x + \frac{1}{4} = \frac{46}{3}x - \frac{6}{5}\)
Step 3: Isolate x
- Rearranging gives:
\(12x - \frac{46}{3}x = -\frac{6}{5} - \frac{1}{4}\)
- Convert to a common denominator to combine fractions (20):
\(-\frac{6}{5} = -\frac{24}{20}, \quad -\frac{1}{4} = -\frac{5}{20}\)
Thus,
\(-\frac{29}{20}\)
Step 4: Solve for x
- Combine the x terms, isolate x, and solve.
This detailed breakdown leads to finding the value of \(x\) through methodical simplification and rearrangement.