find the standard form of a line passing through (-1,4) and (5,8).

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.

Learn how to find the standard form of a line passing through the points (-1, 4) and (5, 8) with a detailed step-by-step solution. This problem covers key concepts in coordinate geometry and linear equations, including the slope formula, point-slope form, and conversion to standard form. Suitable for students in Grades 9-12.

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