Consider the line $y=\frac{4}{5} x-4$
Find the equation of the line that is perpendicular to this line and passes through the point $(4,-4)$. Find the equation of the line that is parallel to this line and passes through the point $(4,-4)$.
Answer
Final Answer: The equation of the line that is perpendicular to the given line and passes through the point \((4,-4)\) is \(\boxed{y = -\frac{5}{4}x + 1}\). The equation of the line that is parallel to the given line and passes through the point \((4,-4)\) is \(\boxed{y = \frac{4}{5}x - 7.2}\).
Step 1 ：The slope of the given line is \(\frac{4}{5}\). The slope of a line perpendicular to this line would be the negative reciprocal of this slope, which is \(-\frac{5}{4}\). The slope of a line parallel to this line would be the same as the slope of the given line, which is \(\frac{4}{5}\).
Step 2 ：We can use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), to find the equations of the lines. Here, \((x_1, y_1)\) is the point through which the line passes and \(m\) is the slope of the line.
Step 3 ：Let's first find the equation of the line that is perpendicular to the given line and passes through the point \((4,-4)\). The y-intercept of the line that is perpendicular to the given line and passes through the point \((4,-4)\) is 1. So, the equation of this line is \(y = -\frac{5}{4}x + 1\).
Step 4 ：Now, let's find the equation of the line that is parallel to the given line and passes through the point \((4,-4)\). The y-intercept of the line that is parallel to the given line and passes through the point \((4,-4)\) is -7.2. So, the equation of this line is \(y = \frac{4}{5}x - 7.2\).
Step 5 ：Final Answer: The equation of the line that is perpendicular to the given line and passes through the point \((4,-4)\) is \(\boxed{y = -\frac{5}{4}x + 1}\). The equation of the line that is parallel to the given line and passes through the point \((4,-4)\) is \(\boxed{y = \frac{4}{5}x - 7.2}\).
