Perpendicular Line Bfcb9F

1. **State the problem:** We need to find the equation of a line that is perpendicular to the line given by $$-7x - 9y = -35$$ and passes through the point $$(-4, -7)$$. 2. **Rewrite the given line in slope-intercept form:** $$-7x - 9y = -35$$ Add $$7x$$ to both sides: $$-9y = 7x - 35$$ Divide both sides by $$-9$$: $$y = \frac{7x - 35}{-9} = -\frac{7}{9}x + \frac{35}{9}$$ 3. **Identify the slope of the given line:** The slope $$m_1$$ is $$-\frac{7}{9}$$. 4. **Find the slope of the perpendicular line:** The slope $$m_2$$ of a line perpendicular to another with slope $$m_1$$ is the negative reciprocal: $$m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{7}{9}} = \frac{9}{7}$$ 5. **Use point-slope form to find the equation of the perpendicular line:** Point-slope form is: $$y - y_1 = m(x - x_1)$$ Using point $$(-4, -7)$$ and slope $$\frac{9}{7}$$: $$y - (-7) = \frac{9}{7}(x - (-4))$$ Simplify: $$y + 7 = \frac{9}{7}(x + 4)$$ 6. **Expand and simplify:** $$y + 7 = \frac{9}{7}x + \frac{36}{7}$$ Subtract 7 from both sides: $$y = \frac{9}{7}x + \frac{36}{7} - 7$$ Convert 7 to $$\frac{49}{7}$$: $$y = \frac{9}{7}x + \frac{36}{7} - \frac{49}{7} = \frac{9}{7}x - \frac{13}{7}$$ 7. **Final equation:** $$y = \frac{9}{7}x - \frac{13}{7}$$ This is the equation of the line perpendicular to $$-7x - 9y = -35$$ and passing through $$(-4, -7)$$.