Find the perpendicular line of y = 1/3x + 5 that passes through the point (-2, 3).
Understand the Problem
The question is asking us to find the equation of a line that is perpendicular to the given line (y = 1/3x + 5) and passes through the point (-2, 3). To solve it, we'll first determine the slope of the perpendicular line and then use the point-slope form of the equation of a line.
Answer
The equation of the perpendicular 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 equation of the given line is in the slope-intercept form $y = mx + b$, where $m$ is the slope. For the line $y = \frac{1}{3}x + 5$, the slope $m$ is $\frac{1}{3}$.
- Calculate the slope of the perpendicular line
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, if the slope of the given line is $\frac{1}{3}$, the slope of the perpendicular line is:
$$ m_{perpendicular} = -\frac{1}{\frac{1}{3}} = -3 $$
- Use the point-slope form to find the equation
Now we will use the point-slope form of a line, which is given by:
$$ y - y_1 = m(x - x_1) $$
where $(x_1, y_1)$ is a point on the line. Here, the point is $(-2, 3)$, and the slope we found is $-3$. Substituting these values into the equation:
$$ y - 3 = -3(x + 2) $$
- Simplify the equation
Distributing the right side of the equation:
$$ y - 3 = -3x - 6 $$
Now, add 3 to both sides:
$$ y = -3x - 6 + 3 $$
This simplifies to:
$$ y = -3x - 3 $$
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
This equation represents a line with a slope of -3 that intersects the y-axis at -3. Perpendicular lines have slopes that multiply to -1, which confirms that the calculations are correct.
Tips
- A common mistake is forgetting to take the negative reciprocal when calculating the slope of the perpendicular line. Always remember to invert the slope and change the sign.
- Another mistake can be improperly applying the point-slope formula, especially when substituting the point coordinates. Carefully check the coordinates used in the formula.
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