Find and equation for the line (-6,-1) (6,-4)

To find the equation of the line that passes through the points \((-6, -1)\) and \((6, -4)\), we'll follow a similar process as before.

### Step 1: Find the slope \(m\)
The slope \(m\) of the line is given by the formula:

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

Substituting the points \((-6, -1)\) and \((6, -4)\):

\[
m = \frac{-4 - (-1)}{6 - (-6)} = \frac{-4 + 1}{6 + 6} = \frac{-3}{12} = -\frac{1}{4}
\]

So, the slope \(m\) is \(-\frac{1}{4}\).

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

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

We can use either of the given points. Let's use \((-6, -1)\):

\[
y - (-1) = -\frac{1}{4}(x - (-6))
\]

Simplifying:

\[
y + 1 = -\frac{1}{4}(x + 6)
\]

Distribute the slope:

\[
y + 1 = -\frac{1}{4}x - \frac{6}{4}
\]

Simplify further:

\[
y + 1 = -\frac{1}{4}x - \frac{3}{2}
\]

Subtract 1 from both sides:

\[
y = -\frac{1}{4}x - \frac{3}{2} - 1
\]

\[
y = -\frac{1}{4}x - \frac{5}{2}
\]

### Final Answer
The equation of the line is:

\[
y = -\frac{1}{4}x - \frac{5}{2}
\]

Would you like any further details or have any questions?

Here are five related questions to expand your understanding:
1. How can you verify that the two points lie on the line you just found?
2. What does a negative slope indicate about the direction of the line?
3. How would you convert this equation to the standard form \(Ax + By = C\)?
4. How can you determine the y-intercept of the line directly from the equation?
5. How would the equation change if one of the points were moved vertically or horizontally?

**Tip:** Always simplify fractions carefully when calculating the slope, as small mistakes can lead to incorrect results in the final equation.

Learn how to find the equation of a line passing through the points (-6, -1) and (6, -4) using the slope formula and point-slope form. This problem involves calculating the slope and applying it to derive the line equation step-by-step. Suitable for students in Grades 7-9.

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