91Ó°ÊÓ

To determine the equation of a line, one of the first steps involves calculating its slope. The slope indicates how steep the line is and the direction it moves. It is calculated using two points on the line. In this exercise, the points (-1, 1) and (5, -3) are used. The formula for finding the slope \( m \) is given by:

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

Here, \( (x_1, y_1) = (-1, 1) \) and \( (x_2, y_2) = (5, -3) \). Substituting these points into the formula gives:

• \( m = \frac{-3 - 1}{5 - (-1)} = \frac{-4}{6} = -\frac{2}{3} \).

This result indicates that for every 3 units the line moves horizontally, it moves 2 units downward. Understanding slope helps in predicting the direction and steepness of the line for any use, from graphing to real-world applications.

Once the slope \( m \) is known, we can proceed to write the equation of the line using the point-slope form. This form is particularly useful for building the equation incrementally, especially when we have one point on the line and the slope. The point-slope form is defined as:

• \( y - y_1 = m(x - x_1) \)

For our exercise, using the slope \( m = -\frac{2}{3} \) and one of the points, say \((-1, 1)\), we substitute

• \( y - 1 = -\frac{2}{3}(x + 1) \)

This equation reflects how the line changes based on its slope from the point \((-1, 1)\). Transforming equations using the point-slope form is straightforward and lays the groundwork for further simplifications to other forms of line equations, making it a powerful tool in algebra.

The slope-intercept form is an efficient way to express a linear equation, providing a clear image of the line's behavior at a glance. Once you have your equation in the point-slope form, the next step is to simplify it into slope-intercept form. This form is represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We simplify the point-slope form equation:

• \( y - 1 = -\frac{2}{3}x - \frac{2}{3} \)

Adding 1 to both sides to isolate \( y \) gives:

• \( y = -\frac{2}{3}x - \frac{2}{3} + 1 \)

Converting \( -\frac{2}{3} + 1 \) to a single fraction results in \( \frac{1}{3} \), so the equation becomes:

• \( y = -\frac{2}{3}x + \frac{1}{3} \)

This form clearly shows the slope \(-\frac{2}{3}\) and the y-intercept \(\frac{1}{3}\), making it ideal for graphing and analyzing linear relationships in a visually intuitive way.