Question

StartFraction 6 x Over 5 EndFraction minus x equals StartFraction x Over 15 EndFraction minus eight thirds

Answer 1

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:

1. \( 15 \cdot \frac{6x}{5} = 3 \cdot 6x = 18x \)
2. \( 15 \cdot x = 15x \)
3. \( 15 \cdot \frac{x}{15} = x \)
4. \( 15 \cdot \frac{8}{3} = 5 \cdot 8 = 40 \)

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 \(x = -20\).