Write two equations, one with a parallel slope and one with a perpendicular slope.
Understand the Problem
The question asks for two equations relative to an initial equation, one with a parallel slope and one with a perpendicular slope. A line is parallel if it has the same slope, it is perpendicular if the slope is the negative reciprocal.
Answer
Parallel: $y = 3x - 5$ Perpendicular: $y = -\frac{1}{3}x - \frac{5}{3}$
Answer for screen readers
The equation of the line parallel to $y = 3x - 5$ and passing through $(1, -2)$ is $y = 3x - 5$. The equation of the line perpendicular to $y = 3x - 5$ and passing through $(1, -2)$ is $y = -\frac{1}{3}x - \frac{5}{3}$.
Steps to Solve
- Identify the slope of the given line
The given line is in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In the equation $y = 3x - 5$, the slope is $3$.
- Determine the slope of the parallel line
Parallel lines have the same slope. Therefore, the slope of the line parallel to $y = 3x - 5$ is also $3$.
- Determine the slope of the perpendicular line
Perpendicular lines have slopes that are negative reciprocals of each other. If the slope of the given line is $3$, then the slope of the perpendicular line is $-\frac{1}{3}$.
- Use the point-slope form to find the equation of the parallel line
The point-slope form of a line is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. We are given the point $(1, -2)$ and the slope $3$. Plugging these values into the point-slope form, we get:
$y - (-2) = 3(x - 1)$
Simplify to slope-intercept form: $y + 2 = 3x - 3$ $y = 3x - 5$
- Use the point-slope form to find the equation of the perpendicular line
Using the point-slope form $y - y_1 = m(x - x_1)$ with the point $(1, -2)$ and the slope $-\frac{1}{3}$, we get:
$y - (-2) = -\frac{1}{3}(x - 1)$
Simplify to slope-intercept form: $y + 2 = -\frac{1}{3}x + \frac{1}{3}$ $y = -\frac{1}{3}x + \frac{1}{3} - 2$ $y = -\frac{1}{3}x + \frac{1}{3} - \frac{6}{3}$ $y = -\frac{1}{3}x - \frac{5}{3}$
The equation of the line parallel to $y = 3x - 5$ and passing through $(1, -2)$ is $y = 3x - 5$. The equation of the line perpendicular to $y = 3x - 5$ and passing through $(1, -2)$ is $y = -\frac{1}{3}x - \frac{5}{3}$.
More Information
The parallel line has the same equation as the original line, $y = 3x - 5$. This indicates that the point $(1, -2)$ already lies on the original line. Substitute the point $(1, -2)$ into $y = 3x - 5$ to verify: $-2 = 3(1) - 5 = -2$.
Tips
A common mistake is not taking the negative reciprocal when finding the slope of the perpendicular line. Remember to flip the fraction and change the sign.
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