Balance the Fraction Equation: Solve for X in -1/7 + 3/4x = 1/8x + 3/14

To solve the equation $-\frac{1}{7} + \frac{3}{4}x = \frac{1}{8}x + \frac{3}{14}$, we complete the following steps:

Step 1: Isolate the terms involving $x$ on one side of the equation and the constant terms on the other side.

Start by subtracting $\frac{1}{8}x$ from both sides:

$-\frac{1}{7} + \frac{3}{4}x - \frac{1}{8}x = \frac{3}{14}$

Step 2: Move the constant term $-\frac{1}{7}$ to the other side:

$\frac{3}{4}x - \frac{1}{8}x = \frac{3}{14} + \frac{1}{7}$

Step 3: Find a common denominator for combining like terms.

For the left side, convert the fractions with denominators 4 and 8 to a common denominator of 8:

$\frac{3}{4}x = \frac{6}{8}x$

So, $\frac{6}{8}x - \frac{1}{8}x = \frac{5}{8}x$

Now consider the right side by converting the fractions with denominators 14 and 7 to a common denominator of 14:

$\frac{1}{7} = \frac{2}{14}$

Therefore, $\frac{3}{14} + \frac{2}{14} = \frac{5}{14}$

Step 4: Equate the simplified terms:

$\frac{5}{8}x = \frac{5}{14}$

Step 5: Solve for $x$ by isolating it using multiplication:

Multiply both sides by $\frac{8}{5}$ to clear the fractional coefficient of $x$:

$x = \frac{5}{14} \times \frac{8}{5}$

Simplify this expression:

$x = \frac{5 \times 8}{14 \times 5} = \frac{8}{14}$

Therefore, the solution to the equation is $x = \frac{8}{14}$.