To solve this problem, we need to determine the equation of the line passing through the points $(3, 4)$ and $(1, -8)$ and then find its $y$-intercept.

Step 1: Find the Slope (Gradient) of the Line

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $(3, 4)$ and $(1, -8)$: $m = \frac{-8 - 4}{1 - 3} = \frac{-12}{-2} = 6$

Step 2: Use the Point-Slope Form to Find the Equation of the Line

The equation of a line with slope $m$ passing through a point $(x_1, y_1)$ is: $y - y_1 = m(x - x_1)$ Using point $(3, 4)$ and $m = 6$: $y - 4 = 6(x - 3)$ Expanding this: $y - 4 = 6x - 18$ $y = 6x - 14$

Step 3: Identify the $y$-Intercept

The $y$-intercept occurs where $x = 0$. In the equation $y = 6x - 14$, substituting $x = 0$ gives: $y = 6(0) - 14 = -14$

Conclusion

The $y$-intercept is $-14$.

Answer

The correct option is: A) -14

Would you like more details on any part of the solution? Here are some related questions to further explore:

1. How do you find the slope of a line given two points?
2. What is the point-slope form of a line, and how is it used?
3. How can you verify the $y$-intercept using different methods?
4. What does the $y$-intercept represent in a linear equation?
5. How would the equation change if we had different points?

Tip: Always substitute the values back into the original points to verify the line equation is correct.