17) through: (1,1) parallel to y=(1)/(6)x-3 A) y=(5)/(6)x+(1)/(6) B) y=(1)/(6)x-(5)/(6) C) y=-(5)/(6)x+(1)/(6) D) y=(1)/(6)x+(5)/(6)

To find the equation of a line that passes through the point $(1,1)$ and is parallel to the line $y=\frac{1}{6}x-3$, we need to use the fact that parallel lines have the same slope.<br /><br />The slope of the given line $y=\frac{1}{6}x-3$ is $\frac{1}{6}$. Since the line we are looking for is parallel to this line, it will also have a slope of $\frac{1}{6}$.<br /><br />Now, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:<br /><br />$y - y_1 = m(x - x_1)$<br /><br />where $(x_1, y_1)$ is a point on the line and $m$ is the slope.<br /><br />Substituting the given point $(1,1)$ and the slope $\frac{1}{6}$ into the point-slope form, we get:<br /><br />$y - 1 = \frac{1}{6}(x - 1)$<br /><br />Simplifying this equation, we have:<br /><br />$y - 1 = \frac{1}{6}x - \frac{1}{6}$<br /><br />Adding 1 to both sides, we get:<br /><br />$y = \frac{1}{6}x + \frac{5}{6}$<br /><br />Therefore, the equation of the line that passes through the point $(1,1)$ and is parallel to $y=\frac{1}{6}x-3$ is $y=\frac{1}{6}x+\frac{5}{6}$.<br /><br />So, the correct answer is D) $y=\frac{1}{6}x+\frac{5}{6}$.