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 $y = -3x - 3$.

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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