Solve for X: Linear Equation with 15/17 and 16/51 Fractions

To solve this equation, we will simplify both sides by combining like terms and then solve for $x$.

Let's break it down step-by-step:

• Simplify the left side of the equation:

The left side is $16 - \frac{15}{17}x + \frac{6}{34}x$. Start by simplifying $\frac{6}{34}x$ to $\frac{3}{17}x$ since $\frac{6}{34} = \frac{3}{17}$. Now, the equation is:

$16 - \frac{15}{17}x + \frac{3}{17}x = 16 - \frac{12}{17}x$.

• Simplify the right side of the equation:

The right side is $18 - 1 - \frac{16}{51}x = 17 - \frac{16}{51}x$.

• Set the equation:

The equation now is:

$16 - \frac{12}{17}x = 17 - \frac{16}{51}x$.

• Move all terms involving $x$ to one side, and constants to the other:

Add $\frac{12}{17}x$ to both sides:

$16 = 17 + \frac{12}{17}x - \frac{16}{51}x$.

• Combine the fractions involving $x$:

The common denominator between $17$ and $51$ is $51$. Convert $\frac{12}{17}x$ to $\frac{36}{51}x$. The equation becomes:

$16 = 17 + \left( \frac{36}{51} - \frac{16}{51} \right)x = 17 + \frac{20}{51}x$.

• Solve for $x$:

Subtract $17$ from both sides:

$16 - 17 = \frac{20}{51}x$, which simplifies to $-1 = \frac{20}{51}x$.

Multiply both sides by the reciprocal of $\frac{20}{51}$, which is $\frac{51}{20}$:

$x = -1 \times \frac{51}{20} = -\frac{51}{20}$.

Thus, the solution to the problem is $x = -\frac{51}{20}$.

This corresponds to choice $1$.