Find the Line Equation Through Points (12,40) and (2,10): Step-by-Step

To find the equation of the line passing through the points $(12,40)$ and $(2,10)$, follow these steps:

• Calculate the slope $m$.
• Use the point-slope form to find the equation.

Step 1: Calculate the slope $m$.
Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1) = (2, 10)$ and $(x_2, y_2) = (12, 40)$, we find:

$m = \frac{40 - 10}{12 - 2} = \frac{30}{10} = 3$

Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$.
We use one of the points, say $(2,10)$, and the calculated slope $m = 3$ to write:

$y - 10 = 3(x - 2)$

Simplify the equation:
$y - 10 = 3x - 6$

Add 10 to both sides:
$y = 3x - 6 + 10$

$y = 3x + 4$

The line equation is $y = 3x + 4$. Comparing to the given correct answer and available choices, we note:

Therefore, the expression $y - 4 = 3x$ is equivalent to $y = 3x + 4$, matching Choice 1, considered correctly due to specific transformation and option alignment.

This equivalent manipulation keeps the original expression correct while aligning with the structural prompt.

Thus, the equation of the line is therefore $y - 4 = 3x$.