What is an equation of the line that passes through the points (3,6) and (−1,−6)?

To find the equation of the line that passes through the points \((3, 6)\) and \((-1, -6)\), we can use the **point-slope form** of a line's equation. Here's the step-by-step process:

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

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

Plugging in the coordinates \((x_1, y_1) = (3, 6)\) and \((x_2, y_2) = (-1, -6)\):

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

So, the slope \( m \) of the line is 3.

### **Step 2: Use the point-slope form to write the equation**
The point-slope form of a line's equation is:

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

We can use either of the points for \((x_1, y_1)\). Let's use \((3, 6)\):

\[
y - 6 = 3(x - 3)
\]

### **Step 3: Simplify the equation**
Distribute the 3 on the right side:

\[
y - 6 = 3x - 9
\]

Add 6 to both sides to solve for \(y\):

\[
y = 3x - 3
\]

### **Final Answer:**
The equation of the line in slope-intercept form is:

\[
y = 3x - 3
\]

Would you like more details or have any questions?

### **Relative Questions:**
1. How do you convert the point-slope form to the standard form of a line's equation?
2. What is the significance of the slope in the equation of a line?
3. Can you find the equation of a line given a point and a slope?
4. How can you check if a point lies on a given line?
5. What changes in the equation if the line passes through the origin?

### **Tip:**  
Remember, the slope of a line indicates its steepness and direction. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.

This detailed guide explains how to find the equation of a line that passes through the points (3,6) and (−1,−6). Key concepts include calculating the slope, using the point-slope form, and simplifying to the slope-intercept form. Perfect for students in Grades 8-10.

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