91Ó°ÊÓ

The slope is a measure of how steep a line is. It represents the rate at which the line rises or falls as you move along it. To calculate the slope (\( m \)) between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula essentially measures the change in the vertical direction (\( y \)) per unit of change in the horizontal direction (\( x \)). In a simpler sense, the slope tells us how many units a line goes up or down for each unit it moves to the right.
For example, between the points \((-3, 7)\) and \((1, 2)\), the slope calculation would look like this:

• Plug in the values: \( m = \frac{2 - 7}{1 - (-3)} = \frac{-5}{4} \)

This result indicates that for every 4 units you move to the right, the line goes down 5 units.

The equation of a line expresses how the coordinates \((x, y)\) relate to each other on a graph. One common form is the slope-intercept form, given by:

• \( y = mx + b \)

Here, \( m \) is the slope, and \( b \) is the y-intercept, which is where the line crosses the y-axis.
In our specific exercise, after calculating the slope (\(-\frac{5}{4}\)) and rearranging the terms using algebra, we eventually find that \(b\), the y-intercept, is \(\frac{13}{4}\). This indicates that the line crosses the y-axis at \(y = \frac{13}{4}\)

Thus, the final equation of the line is:

• \( y = -\frac{5}{4}x + \frac{13}{4} \)

This equation summarizes the entire line with just two parameters, making it a surprisingly powerful tool!

The point-slope form of the equation of a line allows us to use a known point on the line and its slope to find the full equation of the line. The point-slope form is expressed as:

• \( y - y_1 = m(x - x_1) \)

The idea is to "anchor" the line to a specific point \((x_1, y_1)\) on the graph, and use the slope (\( m \)) to describe how the line moves away from this point.
For example, using the point \((-3, 7)\) and the slope \(-\frac{5}{4}\), we plug into the formula:

• \( y - 7 = -\frac{5}{4}(x + 3) \)

This shows us the relative position of other points on the line, based on their distance from \((-3, 7)\).
Converting from point-slope to slope-intercept form requires a bit of algebraic manipulation, solving for \( y \) to transform into the easier-to-read slope-intercept form like we did in our previous sections.