GRAPHS AND FUNCTIONS
Writing equations of lines parallel and perpendicular to a given...
Consider the line $-7 x+4 y=4$.
Find the equation of the line that is perpendicular to this line and passes through the point $(-8,-1)$.
Find the equation of the line that is parallel to this line and passes through the point $(-8,-1)$.
Equation of perpendicular line:
Equation of parallel line:
Answer
Final Answer: The equation of the line that is perpendicular to the line $-7 x+4 y=4$ and passes through the point $(-8,-1)$ is \(\boxed{0.571x + y = -5.571}\). The equation of the line that is parallel to the line $-7 x+4 y=4$ and passes through the point $(-8,-1)$ is \(\boxed{1.75x - y = -13}\).
Step 1 ：Given the line equation in the form $ax + by = c$, the slope of the line is $-a/b$. For the given line $-7x + 4y = 4$, the slope is $7/4$.
Step 2 ：A line perpendicular to the given line will have a slope that is the negative reciprocal of $7/4$, which is $-4/7$.
Step 3 ：A line parallel to the given line will have the same slope, which is $7/4$.
Step 4 ：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 $(-8, -1)$, and $m$ is the slope.
Step 5 ：For the perpendicular line, substituting $m = -4/7$ into the point-slope form gives the equation $0.571x + y = -5.571$.
Step 6 ：For the parallel line, substituting $m = 7/4$ into the point-slope form gives the equation $1.75x - y = -13$.
Step 7 ：Final Answer: The equation of the line that is perpendicular to the line $-7 x+4 y=4$ and passes through the point $(-8,-1)$ is \(\boxed{0.571x + y = -5.571}\). The equation of the line that is parallel to the line $-7 x+4 y=4$ and passes through the point $(-8,-1)$ is \(\boxed{1.75x - y = -13}\).
