To find the y-intercept of the line that passes through the points $(3, 11)$ and $(5, 7)$, we can follow these steps:

Step 1: Find the slope of the line.

The formula to calculate the slope $m$ of a line passing through 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 $(x_1, y_1) = (3, 11)$ and $(x_2, y_2) = (5, 7)$:

$m = \frac{7 - 11}{5 - 3} = \frac{-4}{2} = -2$

So, the slope of the line is $m = -2$.

Step 2: Use the point-slope form to find the y-intercept.

The point-slope form of a line is:

$y - y_1 = m(x - x_1)$

We can use one of the points (let's use $(3, 11)$) and the slope $m = -2$ to find the equation of the line. Substituting these values into the equation:

$y - 11 = -2(x - 3)$

Simplifying:

$y - 11 = -2x + 6$

Now, solve for $y$:

$y = -2x + 6 + 11$

$y = -2x + 17$

Step 3: Identify the y-intercept.

The equation of the line is $y = -2x + 17$. The y-intercept occurs where $x = 0$, so:

$y = -2(0) + 17 = 17$

Thus, the y-intercept of the line is 17.

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Would you like a further breakdown or have any questions? Here are some related questions for you:

1. How do you interpret the slope and y-intercept of a line geometrically?
2. How would you find the equation of a line given two points in three-dimensional space?
3. What happens to the slope if the two points you are given are vertical?
4. How can you determine the slope of a horizontal line?
5. What is the relationship between the slope and parallel or perpendicular lines?

Tip: When finding the y-intercept, always make sure that $x = 0$ in the equation of the line!