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Calculating the slope is a crucial first step in finding the equation of a line that passes through two points. The slope, represented by the letter \( m \), tells you how steep the line is. It's like a 'rise over run', showing how much the line rises vertically compared to how much it runs horizontally across the graph.

To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula finds the difference in the \( y \)-values divided by the difference in the \( x \)-values. It's an easy way to measure how slanted the line is between two points.

For example, when working with the points \((3, -2)\) and \((-1, 4)\), you substitute into the formula to find the slope:

• \[ m = \frac{4 - (-2)}{-1 - 3} = \frac{6}{-4} = -\frac{3}{2} \]

This tells us that for every 2 units we move to the right on the \( x \)-axis, the line goes down 3 units on the \( y \)-axis, giving a slope of \(-\frac{3}{2}\).

Once you have the slope, the next step often involves using the point-slope form of a line's equation to directly plug in the slope and one of the given points. The point-slope form is especially useful when you know a point on the line and its slope.

The point-slope formula is:

• \[ y - y_1 = m(x - x_1) \]

Here, \((x_1, y_1)\) represents any point on the line, and \( m \) is the slope we calculated.

Using the earlier point \((3, -2)\) with our slope \(-\frac{3}{2}\), we substitute these into the formula:

• \[ y + 2 = -\frac{3}{2}(x - 3) \]

This equation expresses a line in terms of its change in \( x \) and \( y \) given a specific point. It's particularly straightforward since it only requires one substitution of point coordinates.
This form is flexible and is often used as a stepping stone to more familiar forms like the standard form or the slope-intercept form.

Finally, once we've used the point-slope form, we can transform the line's equation into the standard form, which is a classic format used in many applications. The standard form is written as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers.

Starting from the slope-intercept form \( y = -\frac{3}{2}x + \frac{5}{2} \), we need to eliminate fractions and rearrange terms.

Here’s a step-by-step transformation:

• Multiply all terms by 2 to get rid of the fractions: \( 2y = -3x + 5 \)
• Rearrange to bring terms on one side: \( 3x + 2y = 5 \)

Now you have the standard form: \( 3x + 2y = 5 \). This form is useful for quickly identifying the x and y-intercepts and is preferred in algebraic solutions because of its uniformity.

In this form, the coefficients \( A \), \( B \), and \( C \) are integers, which makes it ideal for applications that require integer solutions or comparisons.