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Consider the line y=(7)/(3)x+4. Find the equation of the line that is perpendicular to this line and passes through the point (-7,-5). Find the equation of the line that is parallel to this line and passes through the point (-7,-5).

Consider the line y=73x+4 y=\frac{7}{3} x+4 .\newlineFind the equation of the line that is perpendicular to this line and passes through the point (−7,−5) (-7,-5) .\newlineFind the equation of the line that is parallel to this line and passes through the point (−7,−5) (-7,-5) .

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Q. Consider the line y=73x+4 y=\frac{7}{3} x+4 .\newlineFind the equation of the line that is perpendicular to this line and passes through the point (−7,−5) (-7,-5) .\newlineFind the equation of the line that is parallel to this line and passes through the point (−7,−5) (-7,-5) .
  1. Identify slope: Step 11: Identify the slope of the given line.\newlineThe equation of the line is y=73x+4y = \frac{7}{3}x + 4. This is in the slope-intercept form y=mx+by = mx + b, where mm is the slope.\newlineSlope of the given line = 73\frac{7}{3}.
  2. Find perpendicular slope: Step 22: Find the slope of the line perpendicular to the given line.\newlineThe slope of lines that are perpendicular to each other are negative reciprocals. So, the slope of the perpendicular line =−1(73)=−37= -\frac{1}{\left(\frac{7}{3}\right)} = -\frac{3}{7}.
  3. Use point-slope form: Step 33: Use the point-slope form to find the equation of the perpendicular line.\newlineThe point given is (−7,−5)(-7, -5). Using the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope and (x1,y1)(x_1, y_1) is the point,\newliney−(−5)=−37(x−(−7))y - (-5) = -\frac{3}{7}(x - (-7))\newliney+5=−37(x+7)y + 5 = -\frac{3}{7}(x + 7).
  4. Simplify perpendicular equation: Step 44: Simplify the equation of the perpendicular line.\newliney+5=−37x−3y + 5 = -\frac{3}{7}x - 3\newliney=−37x−8y = -\frac{3}{7}x - 8.

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