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The slope of a line measures how steep it is. It tells us how much the y-value changes for a change in the x-value. For linear equations represented in a graph, the slope is the "rise over run." It's essentially the vertical change over the horizontal change between two points on a line.
To find the slope (\( m \)) of a line given two points (\( (x_1, y_1) \) and \( (x_2, y_2) \)), you use the formula:

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

The numerator (\( y_2 - y_1 \)) represents how much the y-values change between the two points, and the denominator (\( x_2 - x_1 \)) represents the change in the x-values.
In our example, points were \( (0, 8) \) and \( (-4, 0) \), which resulted in a slope of 2. This means for every time x increases by 1, y increases by 2.

The slope-intercept form is a way of writing linear equations. This form makes it simple to identify the slope and the y-intercept of a line. It’s given by the equation:

• \( y = mx + b \)

In this equation, \( m \) represents the slope, and \( b \) represents the y-intercept.
The slope-intercept form is very practical. It directly shows you both the steepness of the line and where it crosses the y-axis.
For example, with a slope of 2 and a y-intercept of 8, our equation becomes \( y = 2x + 8 \). This gives us a clear picture of the line on the graph and allows us to calculate y for any value of x quickly.

The y-intercept of a line is the point where the line crosses the y-axis. This is a crucial part of the slope-intercept form equation \( y = mx + b \).
In this equation, \( b \) stands for the y-intercept.
The y-intercept gives us the starting point of the line on the graph when x equals 0.
From the original exercise, substituting in the point \( (0, 8) \) and slope \( m = 2 \) into the equation told us that \( b = 8 \).
This means the line crosses the y-axis at 8. Knowing the y-intercept makes it easy to graph the line, as you can start at the y-intercept and use the slope to determine other points on the line.