To find the standard form of the equation of the line passing through the points $(-1, 4)$ and $(5, 8)$, we will follow these steps:

1. Find the slope of the line.

The slope $m$ of the line passing through the 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 given points $(-1, 4)$ and $(5, 8)$: $m = \frac{8 - 4}{5 - (-1)} = \frac{4}{5 + 1} = \frac{4}{6} = \frac{2}{3}$

2. Use the point-slope form of the line equation.

The point-slope form of a line's equation is: $y - y_1 = m(x - x_1)$ Using the slope $m = \frac{2}{3}$ and the point $(-1, 4)$: $y - 4 = \frac{2}{3}(x + 1)$

3. Convert to slope-intercept form.

Expand and simplify the equation: $y - 4 = \frac{2}{3}x + \frac{2}{3}$ $y = \frac{2}{3}x + \frac{2}{3} + 4$ Convert 4 to a fraction with a denominator of 3: $y = \frac{2}{3}x + \frac{2}{3} + \frac{12}{3} = \frac{2}{3}x + \frac{14}{3}$ Now, multiply the entire equation by 3 to eliminate the fraction: $3y = 2x + 14$

4. Convert to standard form.

Rearrange the equation to get the standard form $Ax + By = C$: $2x - 3y = -14$

Conclusion

The standard form of the equation of the line passing through the points $(-1, 4)$ and $(5, 8)$ is: $2x - 3y = -14$

Do you have any further questions or would you like more details on this? Here are 5 related questions:

1. How do you find the slope of a line if two points on the line are given?
2. What is the difference between standard form and slope-intercept form of a line?
3. How can you determine if two lines are parallel or perpendicular by their equations?
4. How do you find the equation of a line perpendicular to a given line and passing through a specific point?
5. How can the distance from a point to a line be calculated using the standard form of the line?

Tip: When converting a linear equation to standard form, make sure the coefficients of $x$ and $y$ are integers, and the coefficient of $x$ should typically be positive.