The problem asks us to find the equation of the line passing through the points $(-1, 4)$ and $(5, -3)$.

1. Problem Description

The problem asks us to find the equation of the line passing through the points $(-1, 4)$ and $(5, -3)$.

2. Solution Steps

First, we need to find the slope $m$ of the line.

The slope formula is given by:

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

Here, $(x_1, y_1) = (-1, 4)$ and $(x_2, y_2) = (5, -3)$.

Plugging in the values, we get:

$m = \frac{-3 - 4}{5 - (-1)} = \frac{-7}{6}$

Next, we use the point-slope form of a linear equation, which is:

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

Using the point $(-1, 4)$ and the slope $m = -\frac{7}{6}$, we have:

$y - 4 = -\frac{7}{6}(x - (-1))$

$y - 4 = -\frac{7}{6}(x + 1)$

Now, we convert this to slope-intercept form ($y = mx + b$):

$y - 4 = -\frac{7}{6}x - \frac{7}{6}$

$y = -\frac{7}{6}x - \frac{7}{6} + 4$

To add the numbers, we need a common denominator: $4 = \frac{24}{6}$

$y = -\frac{7}{6}x - \frac{7}{6} + \frac{24}{6}$

$y = -\frac{7}{6}x + \frac{17}{6}$

The equation of the line is $y = -\frac{7}{6}x + \frac{17}{6}$.

We can also express the equation in the standard form $Ax + By = C$.

Multiply by 6 to eliminate the fractions:

$6y = -7x + 17$

Rearrange the equation:

$7x + 6y = 17$

3. Final Answer

The equation of the line is $y = -\frac{7}{6}x + \frac{17}{6}$ or $7x + 6y = 17$.