Consider the line \( y=\frac{3}{7} x+6 \). Find the equation of the line that is parallel to this line and passes through the point \( (-9,-5) \). Find the equation of the line that is perpendicular to this line and passes through the point \( (-9,-5) \). Note that the ALEKS graphing calculator may be helpful in checking your answer. Equation of parallel line: Equation of perpendicular line:
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To find a parallel line, we know it will have the same slope as the original line. The slope from the equation \( y=\frac{3}{7} x+6 \) is \( \frac{3}{7} \). Using the point \( (-9,-5) \), we can apply the point-slope form of a line, \( y - y_1 = m(x - x_1) \): \[ y - (-5) = \frac{3}{7}(x - (-9)) \] This simplifies to: \[ y + 5 = \frac{3}{7}(x + 9) \] Distributing and rearranging gives: \[ y = \frac{3}{7}x + \frac{27}{7} - 5 \] Converting \( -5 \) to a fraction yields \( -\frac{35}{7} \), so: \[ y = \frac{3}{7}x - \frac{8}{7} \] Equation of the parallel line: \( y = \frac{3}{7}x - \frac{8}{7} \) For the perpendicular line, we need the slope to be the negative reciprocal of \( \frac{3}{7} \), which is \( -\frac{7}{3} \). Again, using the point-slope form: \[ y - (-5) = -\frac{7}{3}(x - (-9)) \] This simplifies to: \[ y + 5 = -\frac{7}{3}(x + 9) \] Distributing and rearranging gives: \[ y = -\frac{7}{3}x - 21 + 5 \] Converting \( 5 \) to a fraction yields \( \frac{15}{3} \), so: \[ y = -\frac{7}{3}x - \frac{21}{3} + \frac{15}{3} \] Thus, we have: \[ y = -\frac{7}{3}x - \frac{6}{3} \] This simplifies to: \[ y = -\frac{7}{3}x - 2 \] Equation of the perpendicular line: \( y = -\frac{7}{3}x - 2 \)
