We are tasked with finding the equation of the line in point-slope form passing through the points $(2,3)$ and $(3,5)$.

Step 1: Find the slope ($m$) of the line

The formula for slope is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the given points $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (3, 5)$: $m = \frac{5 - 3}{3 - 2} = \frac{2}{1} = 2$

Step 2: Write the equation in point-slope form

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Using the slope $m = 2$ and the point $(2, 3)$: $y - 3 = 2(x - 2)$

Final Answer:

The equation of the line in point-slope form is: $y - 3 = 2(x - 2)$

If you prefer, you can also use the other point $(3, 5)$ to get: $y - 5 = 2(x - 3)$ Both equations are equivalent.

Would you like further clarification on this? Here are some related questions to expand on this concept:

1. What is the slope-intercept form of the line passing through these points?
2. How can you convert a point-slope equation to standard form?
3. Can we generalize the process to find equations for vertical or horizontal lines?
4. What happens when the two given points are the same (coincident points)?
5. How does the point-slope formula relate to the concept of linear functions?

Tip: Always verify your slope calculation by rechecking the difference in $y$-values and $x$-values carefully!