Question: Use the given conditions to write an equation for the line in point-slope form and general form.
Passing through $(5, -5)$ and perpendicular to the line whose equation is $$x - 6y - 7 = 0$$
Part 1 of 2
The equation of the line in point-slope form is
1. **State the problem:** We need to find the equation of a line passing through the point $(5, -5)$ and perpendicular to the line given by $$x - 6y - 7 = 0$$.
2. **Find the slope of the given line:** Rewrite the given line in slope-intercept form $y = mx + b$.
$$x - 6y - 7 = 0 \implies -6y = -x + 7 \implies y = \frac{-x + 7}{-6} = \frac{1}{6}x - \frac{7}{6}$$
So, the slope of the given line is $$m = \frac{1}{6}$$.
3. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal.
$$m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{1}{6}} = -6$$
4. **Write the point-slope form:** The point-slope form is
$$y - y_1 = m(x - x_1)$$
Using point $(5, -5)$ and slope $-6$:
$$y - (-5) = -6(x - 5)$$
Simplify:
$$y + 5 = -6(x - 5)$$
This is the equation in point-slope form.
**Final answer:**
$$\boxed{y + 5 = -6(x - 5)}$$