Find the perpendicular line of y = -2x + 5 that passes through the point (4, -10).
Understand the Problem
The question is asking to find the equation of a line that is perpendicular to the given line (y = -2x + 5) and passes through a specific point (4, -10). To find the perpendicular line, we need to first determine the slope of the given line, then find the negative reciprocal of that slope for the perpendicular line, and use the point-slope form of the line's equation for the line that passes through the given point.
Answer
The equation of the line is $y = \frac{1}{2}x - 12$.
Answer for screen readers
The equation of the line that is perpendicular to $y = -2x + 5$ and passes through the point $(4, -10)$ is
$$ y = \frac{1}{2}x - 12 $$.
Steps to Solve
- Find the slope of the given line
The slope of the given line $y = -2x + 5$ is the coefficient of $x$, which is $-2$.
- Calculate the slope of the perpendicular line
To find the slope of the line that is perpendicular, take the negative reciprocal of the slope of the original line. The negative reciprocal of $-2$ is:
$$ m_{perpendicular} = -\frac{1}{-2} = \frac{1}{2} $$
- Use the point-slope form to write the equation of the new line
The point-slope form of a line is given by the formula:
$$ y - y_1 = m(x - x_1) $$
Where $(x_1, y_1)$ is the point through which the line passes. Here, use the point $(4, -10)$ and the slope $m = \frac{1}{2}$.
- Substitute values into the equation
Plugging in the values into the point-slope form:
$$ y - (-10) = \frac{1}{2}(x - 4) $$
- Simplify the equation
Now simplify the equation:
$$ y + 10 = \frac{1}{2}x - 2 $$
Subtract $10$ from both sides:
$$ y = \frac{1}{2}x - 2 - 10 $$
Thus,
$$ y = \frac{1}{2}x - 12 $$
The equation of the line that is perpendicular to $y = -2x + 5$ and passes through the point $(4, -10)$ is
$$ y = \frac{1}{2}x - 12 $$.
More Information
The resulting equation represents a straight line that has a slope of $\frac{1}{2}$ and intersects the $y$-axis at $-12$. This means for every 2 units you move to the right along the x-axis, the line moves up 1 unit on the y-axis.
Tips
- Forgetting to take the negative reciprocal of the slope when finding the perpendicular slope. Always ensure to change the sign and flip the fraction.
- Errors in plugging values into the point-slope equation can lead to an incorrect equation. Double-check your substitutions.
