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

  1. Identify the slope of the given line

The given line is $y = \frac{1}{3}x + 5$. The slope of this line is $\frac{1}{3}$.

  1. 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:

$$ m = -\frac{1}{\frac{1}{3}} = -3 $$

  1. 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:

$$ y - 3 = -3(x + 2) $$

  1. 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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