91Ó°ÊÓ

To understand the slope of a line, think of it as a measure of how steep the line is. The slope is a ratio comparing how much the y-coordinate (vertical change) changes per unit of change in the x-coordinate (horizontal change). The formula to calculate the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) represents the slope.

In our example, the points are \( (4, 8) \) and \( (10, 0) \). When we substitute these points into the formula, we get: \[ m = \frac{0 - 8}{10 - 4} = \frac{-8}{6} = -\frac{4}{3} \] This negative slope indicates that the line is slanting downwards.

The point-slope form of a linear equation is useful when you know the slope of a line and one point on the line. The general formula is: \[ y - y_1 = m(x - x_1) \] Here, \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope.

In our case, we use the point \( (4, 8) \) and the slope \( m = -\frac{4}{3} \). Plugging these values into the point-slope form, we get: \[ y - 8 = -\frac{4}{3}(x - 4) \] This equation expresses the line in point-slope form, ready for conversion into another form if needed.

Once we have the equation in point-slope form, the next step is to convert it into slope-intercept form, \( y = mx + b \). This format is often more useful because it clearly shows the slope \( m \) and the y-intercept \( b \) of the line.
Starting from our point-slope form equation: \[ y - 8 = -\frac{4}{3}(x - 4) \]
First, distribute the slope on the right-hand side: \[ y - 8 = -\frac{4}{3}x + \frac{16}{3} \]
Next, isolate \( y \) on one side of the equation by adding 8 to both sides: \[ y = -\frac{4}{3}x + \frac{16}{3} + 8 \]
Convert 8 to a fraction with the same denominator (3) and combine like terms: \[ y = -\frac{4}{3}x + \frac{16}{3} + \frac{24}{3} = -\frac{4}{3}x + \frac{40}{3} \]
Now, the equation is in slope-intercept form, \( y = -\frac{4}{3}x + \frac{40}{3} \), making it easier to understand the line's slope and y-intercept.