Find the general form of the equation of the line passing through the point (3*, −*1) and

perpendicular to the line *x* + 3*y −* 7 = 0.

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

### Step 1: Find the slope of the given line
The given line equation is:
\[
x + 3y - 7 = 0
\]
Rewrite this equation in the slope-intercept form \( y = mx + b \) to find the slope:
\[
3y = -x + 7
\]
\[
y = -\frac{1}{3}x + \frac{7}{3}
\]
Thus, the slope of the given line is \( m_1 = -\frac{1}{3} \).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, if the slope of the given line is \( m_1 = -\frac{1}{3} \), then the slope of the perpendicular line, \( m_2 \), is:
\[
m_2 = 3
\]

### Step 3: Use the point-slope form of the line equation
The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
Here, \( m = 3 \), and the point \( (x_1, y_1) = (3, -1) \).

Substitute these values into the point-slope form:
\[
y - (-1) = 3(x - 3)
\]
\[
y + 1 = 3(x - 3)
\]
\[
y + 1 = 3x - 9
\]
\[
y = 3x - 10
\]

### Step 4: Write the equation in general form
The general form of a line equation is \( Ax + By + C = 0 \). To convert \( y = 3x - 10 \) into this form:
\[
3x - y - 10 = 0
\]

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

### Final Answer:
The equation of the line perpendicular to \( x + 3y - 7 = 0 \) and passing through the point \( (3, -1) \) is:
\[
3x - y - 10 = 0
\]

---

Would you like any further explanation or details on any of the steps?

Here are some related questions for further practice:
1. What is the slope of a line parallel to \( x + 3y - 7 = 0 \)?
2. How do you find the equation of a line parallel to \( x + 3y - 7 = 0 \) and passing through \( (4, 2) \)?
3. What is the equation of a line perpendicular to \( 2x - 5y + 3 = 0 \)?
4. How do you find the intersection point of two lines?
5. What is the distance from the point \( (3, -1) \) to the line \( x + 3y - 7 = 0 \)?

**Tip:** The negative reciprocal of the slope of a line gives the slope of any line perpendicular to it.

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 general form of the equation of a line passing through the point (3, -1) and perpendicular to the line x + 3y - 7 = 0. This step-by-step guide covers concepts in coordinate geometry, slope, and perpendicularity, suitable for students in Grades 9-11.

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