To solve the problem of finding the equation of the line that passes through the points $(2, -1)$ and $(4, 7)$ in slope-intercept form, we can follow these steps:

Step 1: Calculate the Slope (m)

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Substitute the coordinates $(x_1, y_1) = (2, -1)$ and $(x_2, y_2) = (4, 7)$:

$m = \frac{7 - (-1)}{4 - 2} = \frac{7 + 1}{4 - 2} = \frac{8}{2} = 4$

So, the slope $m$ is 4.

Step 2: Use the Point-Slope Form to Find the Equation

The point-slope form of the equation of a line is:

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

Using the point $(2, -1)$ and the slope $m = 4$:

$y - (-1) = 4(x - 2)$

Simplify the equation:

$y + 1 = 4(x - 2)$

$y + 1 = 4x - 8$

Step 3: Solve for y to Get the Equation in Slope-Intercept Form

Now, solve for $y$:

$y = 4x - 8 - 1$

$y = 4x - 9$

Final Answer

The equation of the line in slope-intercept form is:

$\boxed{y = 4x - 9}$

This is the required equation of the line that passes through the points $(2, -1)$ and $(4, 7)$.

Would you like more details or have any questions about this solution?

Related Questions:

1. How do you find the equation of a line given the slope and one point?
2. What is the significance of the slope in the equation of a line?
3. How do you convert from point-slope form to slope-intercept