Solving the Fraction-Based Linear Equation: 1/8x - 3/4 + 1/9 = -1/4 + 3/4x - 1/2x

To solve the given linear equation $\frac{1}{8}x - \frac{3}{4} + \frac{1}{9} = -\frac{2}{8} + \frac{3}{4}x - \frac{1}{2}x$, we need to follow these steps:

First, simplify both sides of the equation:

On the left-hand side, which is $\frac{1}{8}x - \frac{3}{4} + \frac{1}{9}$:

• Simplifying, it remains $\frac{1}{8}x - \frac{3}{4} + \frac{1}{9}$.
• Convert $-\frac{3}{4} + \frac{1}{9}$ to a common denominator. The least common multiple of $4$ and $9$ is $36$.
• $-\frac{3}{4} = -\frac{27}{36}$ and $\frac{1}{9} = \frac{4}{36}$, so $-\frac{27}{36} + \frac{4}{36} = -\frac{23}{36}$.
• The left-hand side is now $\frac{1}{8}x - \frac{23}{36}$.

Now, simplify the right-hand side, which is $-\frac{2}{8} + \frac{3}{4}x - \frac{1}{2}x$:

• $-\frac{2}{8} = -\frac{1}{4}$.
• Simplify $\frac{3}{4}x - \frac{1}{2}x$. The common denominator for $\frac{3}{4}$ and $\frac{1}{2}$ is $4$.
• So $\frac{3}{4}x - \frac{1}{2}x = \frac{3}{4}x - \frac{2}{4}x = \frac{1}{4}x$.
• The right-hand side is now $-\frac{1}{4} + \frac{1}{4}x$.

Combine like terms across the equation:

• We aim to move all terms involving $x$ to one side and constants to the other.
• Subtract $\frac{1}{4}x$ from both sides: $\frac{1}{8}x - \frac{1}{4}x - \frac{23}{36} = -\frac{1}{4}$.
• Bring $-\frac{23}{36}$ to the right side: $\frac{1}{8}x - \frac{1}{4}x = -\frac{1}{4} + \frac{23}{36}$.

Simplify and solve for $x$:

• $\frac{1}{8}x - \frac{1}{4}x = \frac{1}{8}x - \frac{2}{8}x = -\frac{1}{8}x$.
• Add $\frac{23}{36}$ to $-\frac{1}{4}$, by finding a common denominator of $36$.
• $-\frac{1}{4} = -\frac{9}{36}$, so $-\frac{9}{36} + \frac{23}{36} = \frac{14}{36} = \frac{7}{18}$.
• Now we have: $-\frac{1}{8}x = \frac{7}{18}$.
• Multiply both sides by $-8$ to solve for $x$: $(-8) \cdot \left(-\frac{1}{8}x\right) = (-8) \cdot \left(\frac{7}{18}\right)$.
• This simplifies to $x = -\frac{56}{18} = -\frac{28}{9}$.
• $-\frac{28}{9}$ can be rewritten as a mixed number: $-\frac{28}{9} = -3\frac{1}{9}$.

Therefore, the solution is:

$x = -3\frac{1}{9}$.