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 = (1/3)x + 5 and passes through the point (-2, -3). To solve this, we first need to determine the slope of the given line, then find the negative reciprocal of that slope for the perpendicular line, and finally use the point-slope form to derive the equation of the line.
Answer
The equation of the line is $y = -3x - 9$.
Answer for screen readers
The equation of the line that is perpendicular to the given line and passes through the point (-2, -3) is:
$$ y = -3x - 9 $$
Steps to Solve
- Determine the slope of the given line
The slope of the line given by the equation $y = \frac{1}{3}x + 5$ is $\frac{1}{3}$.
- Find 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, the negative reciprocal of $\frac{1}{3}$ is:
$$ m_{\text{perpendicular}} = -\frac{1}{\left(\frac{1}{3}\right)} = -3 $$
- Use the point-slope form of the equation
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 the point $(-2, -3)$ and $m$ is the slope we found in the previous step. Substituting the values:
$$ y - (-3) = -3(x - (-2)) $$
- Simplify the equation
Distributing and simplifying the equation gives:
$$ y + 3 = -3(x + 2) $$
Now, let’s simplify further:
$$ y + 3 = -3x - 6 $$
Subtracting 3 from both sides:
$$ y = -3x - 6 - 3 $$
So, we have:
$$ y = -3x - 9 $$
The equation of the line that is perpendicular to the given line and passes through the point (-2, -3) is:
$$ y = -3x - 9 $$
More Information
This equation represents a line with a slope of -3, which indicates that for every unit increase in $x$, the $y$ decreases by 3 units. The perpendicular line intersects the original line at a right angle, demonstrating an important property within coordinate geometry.
Tips
- Incorrectly calculating the negative reciprocal: It's common to mistakenly find the reciprocal without considering the negative sign. Always remember to change the sign when calculating the perpendicular slope.
- Errors in simplifying equations: Neglecting to correctly distribute or combine like terms can lead to incorrect final equations. Double-check each step thoroughly.
