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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.

Find the y-intercept of the straight line which passes through the points (3, 4) and (1, -8).

Learn how to calculate the y-intercept of a line passing through the points (3, 4) and (1, -8). This problem involves finding the slope of the line and using the point-slope form to determine the equation of the line. Suitable for students in Grades 9-11.

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