To find the equation of the line that passes through the points $(1, 1)$ and $(2, 4)$, we'll use the slope-intercept form, which is given by:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Step 1: Find the slope ($m$).

The formula for the slope 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 given points $(1, 1)$ and $(2, 4)$:

$m = \frac{4 - 1}{2 - 1} = \frac{3}{1} = 3$

Step 2: Use the slope to find the y-intercept ($b$).

Using the slope-intercept form $y = mx + b$, plug in the slope $m = 3$ and one of the points, say $(1, 1)$:

$1 = 3 \cdot 1 + b$

Solving for $b$:

$1 = 3 + b \implies b = 1 - 3 = -2$

Step 3: Write the final equation.

Substituting $m = 3$ and $b = -2$ into the slope-intercept form:

$y = 3x - 2$

So, the equation of the line in slope-intercept form is:

$\boxed{y = 3x - 2}$

Would you like more details, or do you have any questions?

Here are some related questions you might find useful:

1. How do you find the equation of a line given a slope and a point?
2. What is the significance of the y-intercept in the slope-intercept form?
3. How can you convert an equation from slope-intercept form to standard form?
4. How do you determine if two lines are parallel or perpendicular?
5. What is the point-slope form of a line, and when is it useful?

Tip: Always double-check your calculations when finding slopes and intercepts to avoid simple arithmetic errors!