Find the perpendicular line of y = -1/4x - 2 that passes through the point (-1, -6)
Understand the Problem
The question is asking us to find the equation of a line that is perpendicular to the given line (y = -1/4x - 2) and passes through the specified point (-1, -6). To solve this, we need to determine the slope of the given line, find the negative reciprocal of that slope for the perpendicular line, and then use the point-slope form of a linear equation to find the equation of the desired line.
Answer
The equation is $y = 4x - 2$.
Answer for screen readers
The equation of the line that is perpendicular to the given line and passes through the point $(-1, -6)$ is $y = 4x - 2$.
Steps to Solve
- Identify the slope of the given line
The given line is $y = -\frac{1}{4}x - 2$.
From the equation, we can see that the slope ($m$) of this line is $-\frac{1}{4}$.
- Find the slope of the perpendicular line
To find the slope of a line perpendicular to the given line, we take the negative reciprocal of the slope.
The negative reciprocal of $-\frac{1}{4}$ is $\frac{4}{1} = 4$.
- Use the point-slope form of a line
Now that we have the slope of the perpendicular line, which is $4$, and we have the point $(-1, -6)$ through which the line passes, we can use the point-slope form of the linear equation:
$$ y - y_1 = m(x - x_1) $$
Substituting in our values ($m = 4$, $x_1 = -1$, $y_1 = -6$):
$$ y + 6 = 4(x + 1) $$
- Simplify the equation
Now, we simplify the equation to slope-intercept form:
$$ y + 6 = 4x + 4 $$
Subtract $6$ from both sides:
$$ y = 4x - 2 $$
The equation of the line that is perpendicular to the given line and passes through the point $(-1, -6)$ is $y = 4x - 2$.
More Information
This equation represents a line with a slope of $4$ and a y-intercept of $-2$. It shows that for every unit increase in $x$, the value of $y$ increases by $4$.
Tips
- Forgetting to take the negative reciprocal: When finding the slope of the perpendicular line, remember to change the sign and invert the value.
- Mistakes in substituting the point: Make sure to accurately substitute the values of the point in the point-slope formula to avoid errors in the equation.
