 3. 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.

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:
\[
3x - y - 10 = 0
\]

Let me know if you'd like further clarification or details!

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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!

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.

Learn how to find the equation of a line that is perpendicular to x + 3y − 7 and passes through the point (3, −1). This problem covers key geometry concepts, including the slopes of perpendicular lines and using point-slope and slope-intercept forms to derive the general equation. Suitable for students in Grades 9-11.

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