Find the general form equation for the line through the points.
Recall that general form is AX + BY + C = 0
Points: (- 7, - 2) and (1, 6)

To find the general form equation of the line passing through the points \((-7, -2)\) and \( (1, 6)\), follow these steps:

### Step 1: Find the slope of the line
The slope \(m\) of the line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the given points \((-7, -2)\) and \( (1, 6)\):

\[
m = \frac{6 - (-2)}{1 - (-7)} = \frac{6 + 2}{1 + 7} = \frac{8}{8} = 1
\]

### Step 2: Use the point-slope form to find the equation of the line
The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Using the point \((-7, -2)\) and the slope \(m = 1\):

\[
y - (-2) = 1(x - (-7))
\]

Simplifying:

\[
y + 2 = x + 7
\]

### Step 3: Rearrange to get the equation in general form
Move all terms to one side of the equation:

\[
x - y + 5 = 0
\]

### Step 4: Express the equation in general form \(AX + BY + C = 0\)
The equation is:

\[
\boxed{x - y + 5 = 0}
\]

This is the general form equation of the line passing through the given points.

Would you like more details on any step, or do you have any questions?

### Related Questions
1. How do you convert the general form equation to slope-intercept form?
2. What is the significance of the slope in a linear equation?
3. How can you verify if a point lies on a given line using its equation?
4. What are the conditions for two lines to be parallel or perpendicular?
5. How do you determine the x-intercept and y-intercept of a line from its equation?

### Tip
Always check your final equation by substituting the original points to ensure they satisfy the equation.

Learn how to find the general form equation of a line passing through the points (-7, -2) and (1, 6). This problem involves analytical geometry and linear equations, including the slope formula and converting the equation to general form. Suitable for Grades 9-12.

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