To solve the equation
\[ \frac{6x}{5} - x = \frac{x}{15} - \frac{8}{3}, \]
we will first eliminate the fractions by finding a common denominator. The least common multiple of the denominators (5, 15, and 3) is 15.
Now, multiply each term of the equation by 15 to eliminate the fractions:
\[ 15 \left(\frac{6x}{5}\right) - 15x = 15 \left(\frac{x}{15}\right) - 15 \left(\frac{8}{3}\right). \]
Calculating each term:
- \( 15 \cdot \frac{6x}{5} = 3 \cdot 6x = 18x \)
- \( 15 \cdot x = 15x \)
- \( 15 \cdot \frac{x}{15} = x \)
- \( 15 \cdot \frac{8}{3} = 5 \cdot 8 = 40 \)
Now substituting these back in gives:
\[ 18x - 15x = x - 40. \]
Simplifying both sides:
\[ 3x = x - 40. \]
Next, subtract \(x\) from both sides to isolate the \(x\):
\[ 3x - x = -40, \]
which simplifies to:
\[ 2x = -40. \]
Now divide both sides by 2 to find \(x\):
\[ x = -20. \]
Thus, the solution to the equation is
\[ \boxed{-20}. \]
You can check the solution by substituting \(x = -20\) back into the original equation to verify both sides are equal.
Verification:
Left-hand side: \[ \frac{6(-20)}{5} - (-20) = \frac{-120}{5} + 20 = -24 + 20 = -4. \]
Right-hand side: \[ \frac{-20}{15} - \frac{8}{3} = -\frac{4}{3} - \frac{8}{3} = -\frac{12}{3} = -4. \]
Both sides of the equation match, confirming the solution is correct. Thus, the solution is \(x = -20\).
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