Consider the line \( y=6 x-7 \). Find the equation of the line that is parallel to this line and passes through the point \( (8,-3) \). Find the equation of the line that is perpendicular to this line and passes through the point \( (8,-3) \). Note that the ALEKS graphing calculator may be helpful in checking your answer. Equation of parallel line: Equation of perpendicular line: \( y=\frac{1}{\square} \)

To find the equation of the parallel line, we know that it will have the same slope as the original line \( y = 6x - 7 \), which is 6. Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (8, -3) \) and \( m = 6 \): \[ y + 3 = 6(x - 8) \\ y + 3 = 6x - 48 \\ y = 6x - 51 \] So, the equation of the parallel line is \( y = 6x - 51 \). For the perpendicular line, the slope will be the negative reciprocal of 6, which is \( -\frac{1}{6} \). Again, using the point-slope form: \[ y + 3 = -\frac{1}{6}(x - 8) \\ y + 3 = -\frac{1}{6}x + \frac{8}{6} \\ y + 3 = -\frac{1}{6}x + \frac{4}{3} \] To isolate \( y \): \[ y = -\frac{1}{6}x + \frac{4}{3} - 3 \\ y = -\frac{1}{6}x + \frac{4}{3} - \frac{9}{3} \\ y = -\frac{1}{6}x - \frac{5}{3} \] So, the equation of the perpendicular line is \( y = -\frac{1}{6}x - \frac{5}{3} \). Final answer: Equation of parallel line: \( y = 6x - 51 \) Equation of perpendicular line: \( y = -\frac{1}{6}x - \frac{5}{3} \)