To find the general form of the equation of the line passing through the point $(3, -1)$ and perpendicular to the line $x + 3y - 7 = 0$, follow these steps:

Step 1: Find the slope of the given line

First, rearrange the given line equation $x + 3y - 7 = 0$ into slope-intercept form $y = mx + b$, where $m$ is the slope: $x + 3y - 7 = 0$ $3y = -x + 7$ $y = -\frac{1}{3}x + \frac{7}{3}$ The slope of this line is $m = -\frac{1}{3}$.

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Since the slope of the given line is $-\frac{1}{3}$, the slope of the perpendicular line will be: $m_{\perp} = 3$

Step 3: Use the point-slope form of the equation

Now, we can use the point-slope form of the line equation: $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (3, -1)$ and $m = 3$. Substituting these values: $y - (-1) = 3(x - 3)$ $y + 1 = 3(x - 3)$ $y + 1 = 3x - 9$ $y = 3x - 10$

Step 4: Convert to general form

To write the equation in general form, move all terms to one side of the equation: $3x - y - 10 = 0$

Thus, the general form of the equation of the line is: $3x - y - 10 = 0$

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Related Questions:

1. How do you find the slope of a line given two points?
2. How do you convert a line equation from slope-intercept form to general form?
3. How can you verify if two lines are perpendicular?
4. What is the point-slope form of a line, and when is it used?
5. How do you find the equation of a line parallel to a given line?

Tip: Always remember, perpendicular lines have slopes that are negative reciprocals of each other!