To determine whether it's possible to write the equation of the line passing through points $A(1, 2)$, $B(2, 4)$, and $C(a, a + 24)$ for some value of $a$, we need to examine whether these points are collinear. If they are collinear, we can find the equation of the line, and if they are not, it would be impossible.

Step 1: Find the Slope of Line AB

The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Using points $A(1, 2)$ and $B(2, 4)$, $m_{AB} = \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2$ So, the slope of line $AB$ is $2$.

Step 2: Check if Point C Lies on Line AB

For point $C(a, a + 24)$ to lie on the same line as $A$ and $B$, the slope of $AC$ must also be $2$. Therefore, we set up an equation for the slope between points $A(1, 2)$ and $C(a, a + 24)$ to be equal to $2$: $m_{AC} = \frac{(a + 24) - 2}{a - 1} = 2$

Step 3: Solve for $a$

$\frac{a + 24 - 2}{a - 1} = 2$ $\frac{a + 22}{a - 1} = 2$ Now, cross-multiply to solve for $a$: $a + 22 = 2(a - 1)$ $a + 22 = 2a - 2$ $22 + 2 = 2a - a$ [