91Ó°ÊÓ

When you have two points on a line, calculating the slope helps understand how steep the line is. Slope measures the rate at which the line rises or falls as you move along it. To find the slope (often represented by the letter "m"), use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
In this formula, \(x_1\) and \(y_1\) are the coordinates of the first point, while \(x_2\) and \(y_2\) are the coordinates of the second point.
In our problem, the points are \((-2, 8)\) and \((4, 6)\). Substituting these into the formula:

• First, calculate the change in \(y\): \(6 - 8 = -2\).
• Next, find the change in \(x\): \(4 - (-2) = 6\).
• Finally, divide the change in \(y\) by the change in \(x\): \(\frac{-2}{6} = -\frac{1}{3}\).

This means the slope of the line is \(-\frac{1}{3}\), indicating a slight downward trend as you move from left to right.

The point-slope form is a way to write the equation of a line, especially handy when you know a point on the line and its slope. This form uses the equation \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the known point, and \(m\) is the slope.
This method is very useful because it allows you to plug in the information directly into the formula. Let's use the point \((-2, 8)\) that we know is on the line and the slope \(-\frac{1}{3}\) we calculated:

• Start by substituting these values into the point-slope formula: \( y - 8 = -\frac{1}{3}(x + 2) \).

This equation now represents our line with the given slope and passes through the specified point.
Working with the point-slope form first can often make it easier to then convert into simpler forms like the slope-intercept form later.

Another popular way of expressing the equation of a line is the slope-intercept form. This is written as \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis.
To convert from the point-slope form \( y - 8 = -\frac{1}{3}(x + 2) \) to the slope-intercept form, you will expand and simplify:

• Distribute the slope \(-\frac{1}{3}\) across \( (x + 2) \): \( y - 8 = -\frac{1}{3}x - \frac{2}{3} \).
• Add 8 to both sides to isolate \(y\):
  \( y = -\frac{1}{3}x - \frac{2}{3} + 8 \).
• Convert 8 to \( \frac{24}{3} \) to add it to \( -\frac{2}{3} \):
  \( y = -\frac{1}{3}x + \frac{22}{3} \).

The final equation \( y = -\frac{1}{3}x + \frac{22}{3} \) is the slope-intercept form, clearly showing the slope and y-intercept. This form is great for quickly understanding how the line behaves and where it crosses the y-axis.