To write the equation of the line passing through the points $(7, 8)$ and $(3, 4)$, we can use the point-slope formula or slope-intercept form. Let's follow these steps:

Step 1: Find the slope

The formula for the slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the points $(7, 8)$ and $(3, 4)$:

$m = \frac{4 - 8}{3 - 7} = \frac{-4}{-4} = 1$

So, the slope $m$ is 1.

Step 2: Use the point-slope form

Now that we have the slope $m = 1$, we can use the point-slope form of the equation of a line:

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

Using the point $(7, 8)$ and $m = 1$, we get:

$y - 8 = 1(x - 7)$

Step 3: Simplify the equation

Now, simplify to get the equation in slope-intercept form $y = mx + b$:

$y - 8 = x - 7$

$y = x + 1$

Thus, the equation of the line is:

$y = x + 1$

Final Answer:

The equation of the line passing through the points $(7, 8)$ and $(3, 4)$ is:

$y = x + 1$

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How would the equation change if the slope were different?
2. How can we find the equation if the line passes through a point and has a known slope?
3. What is the significance of the y-intercept in the equation of a line?
4. How do we graph this line on the coordinate plane?
5. What happens if the slope is zero or undefined?

Tip: When you have the slope, always double-check it by plugging the points back into the slope formula to ensure accuracy.