To find the equation of the line passing through two points $(6, -3)$ and $(3, 1)$, we follow these steps:

Step 1: Find the slope ($m$)

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points: $m = \frac{1 - (-3)}{3 - 6} = \frac{1 + 3}{-3} = \frac{4}{-3} = -\frac{4}{3}$

Step 2: Use the point-slope form

The point-slope form of a line is: $y - y_1 = m(x - x_1)$

Using the point $(6, -3)$ and $m = -\frac{4}{3}$, we get: $y - (-3) = -\frac{4}{3}(x - 6)$ $y + 3 = -\frac{4}{3}(x - 6)$

Step 3: Simplify to slope-intercept form ($y = mx + b$)

Distribute the slope: $y + 3 = -\frac{4}{3}x + \frac{24}{3}$ $y + 3 = -\frac{4}{3}x + 8$

Subtract 3 from both sides: $y = -\frac{4}{3}x + 5$

Final Answer:

The equation of the line is: $y = -\frac{4}{3}x + 5$

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Do you have any questions or would you like to see alternative forms? Here are some related questions:

1. What is the interpretation of the slope and y-intercept in this equation?
2. How can this line be graphed on a Cartesian plane?
3. What is the distance between the two given points?
4. What are the x- and y-intercepts of this line?
5. How can this equation be converted to standard form?

Tip: Always double-check calculations by substituting the points into the final equation to verify correctness!