To find the equation of the line that passes through the points $(3, 4)$ and $(-3, -10)$, we can follow these steps:

Step 1: Find the slope $m$ using the slope formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$ Here, $(x_1, y_1) = (3, 4)$ and $(x_2, y_2) = (-3, -10)$.

Substitute the values: $m = \frac{-10 - 4}{-3 - 3} = \frac{-14}{-6} = \frac{7}{3}$ So, the slope $m = \frac{7}{3}$.

Step 2: Use the point-slope form to write the equation of the line.

The point-slope form is: $y - y_1 = m(x - x_1)$ Using the point $(3, 4)$ and slope $m = \frac{7}{3}$, substitute into the formula: $y - 4 = \frac{7}{3}(x - 3)$

Step 3: Simplify the equation.

Distribute the slope: $y - 4 = \frac{7}{3}x - \frac{7}{3} \cdot 3$ $y - 4 = \frac{7}{3}x - 7$ Add 4 to both sides: $y = \frac{7}{3}x - 7 + 4$ $y = \frac{7}{3}x - 3$

Final Equation:

The equation of the line is: $y = \frac{7}{3}x - 3$

Would you like more details or have any further questions?

Here are five related questions to expand on this:

1. How do you find the equation of a line if you know only one point and the slope?
2. What is the importance of the slope in determining the direction of a line?
3. How can you check if a given point lies on a line using its equation?
4. What is the difference between slope-intercept form and point-slope form of a line?
5. How can you find the x- or y-intercept from the equation of a line?

Tip: Always double-check your slope calculation to avoid sign mistakes when working with negative coordinates.