Find the perpendicular line of y = 1/3x + 5 that passes through the point (-2, 3).
Understand the Problem
The question is asking to find the equation of the line that is perpendicular to the given line y = (1/3)x + 5 and passes through the point (-2, 3). To solve this, we first need to determine the slope of the given line, find the negative reciprocal for the perpendicular slope, and then use point-slope form to find the equation.
Answer
The equation of the line is $y = -3x - 3$.
Answer for screen readers
The equation of the line that is perpendicular to $y = \frac{1}{3}x + 5$ and passes through the point $(-2, 3)$ is
$$ y = -3x - 3 $$
Steps to Solve
- Identify the slope of the given line
The given line is $y = \frac{1}{3}x + 5$. The slope of this line is $\frac{1}{3}$.
- Calculate the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope $m$ of the perpendicular line is given by:
$$ m = -\frac{1}{\frac{1}{3}} = -3 $$
- Use point-slope form to find the equation of the new line
The point-slope form of a line is given by:
$$ y - y_1 = m(x - x_1) $$
Where $(x_1, y_1)$ is the point on the line. Substituting $(-2, 3)$ for $(x_1, y_1)$ and $-3$ for $m$, we get:
$$ y - 3 = -3(x + 2) $$
- Simplify the equation
Now we will simplify this equation:
$$ y - 3 = -3x - 6 $$
Adding 3 to both sides results in:
$$ y = -3x - 3 $$
The equation of the line that is perpendicular to $y = \frac{1}{3}x + 5$ and passes through the point $(-2, 3)$ is
$$ y = -3x - 3 $$
More Information
The slope of the given line was utilized to determine the perpendicular slope, which is essential in geometry for identifying relationships between lines. The point-slope form is a convenient way to create a line equation from a slope and a point.
Tips
- Confusing the slope with the y-intercept: It's easy to mix these up when reading the equation of the line.
- Forgetting to take the negative reciprocal when finding the slope of the perpendicular line.
- Errors in using the point-slope form; ensure to correctly plug in the values for the point.
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