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To understand linear equations, we first need to calculate the slope of a line. The slope is a measure of how steep a line is. It tells us how much y changes for a one-unit change in x.

To calculate the slope (denoted as \( m \)), we use the formula:

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

This formula essentially finds the 'rise' over the 'run', or the vertical change divided by the horizontal change, between two points. Using our example points (2,8) and (5,0), we plug these values into the formula:

• \( m = \frac{0 - 8}{5 - 2} = \frac{-8}{3} \)

So, the slope of the line through these points is \(-8/3\).

Remember that a negative slope means the line is going downwards as you move from left to right.

The point-slope form is a simple way to describe a linear equation. It's particularly useful when you know a point on the line and the slope. The point-slope form is written as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a specific point on the line, and \( m \) is the slope. For our example, we can choose the point (2,8) along with the slope \(-8/3\) that we calculated previously.

This substitution gives us:

• \( y - 8 = -\frac{8}{3}(x - 2) \)

This formula now represents our line, describing its behavior across every point it passes through. While it’s not the most compact form, it is incredibly intuitive for pinpointing the line when used correctly.

Once we have the equation of a line, sketching it can further consolidate understanding. First, consider the y-intercept, which is the point where the line crosses the y-axis. From our simplified equation:

• \( y = -\frac{8}{3}x + \frac{40}{3} \)

We see that the y-intercept is \( \left(0, \frac{40}{3} \right) \), approximately \( 13.33 \). This is an anchor point on the graph.

Next, use the slope to determine the line’s direction. A slope of \(-8/3\) means for every 3 units you move right, you move 8 units down. Starting at (0,40/3), go 3 units to the right (reach x = 3) and 8 units down from about 13.33 to approximately 5.33.

Mark the line passing through your initial points (2,8) and (5,0), confirming the line's path. Sketching reinforces understanding, translating the equation into a visual representation.