7.4 The slope-intercept form of a line

Slope

The slope-intercept form of the equation of a line

The slope is a measure of the steepness of a line

Sample set c

Find the slope and the $y$ -intercept of the following lines.

Practice set c

The formula for the slope of a line

We have observed that the slope is a measure of the steepness of a line. We wish to develop a formula for measuring this steepness.

It seems reasonable to develop a slope formula that produces the following results:

Steepness of line $1>$ steepness of line 2.

Consider a line on which we select any two points. We’ll denote these points with the ordered pairs $(x_{1,}{y}_1)$ and $(x_{2,}{y}_2)$ . The subscripts help us to identify the points.

$(x_{1,}{y}_1)$ is the first point. Subscript 1 indicates the first point.
$(x_{2,}{y}_2)$ is the second point. Subscript 2 indicates the second point.

The difference in $x$ values $(x_2-x_1)$ gives us the horizontal change, and the difference in $y$ values $(y_2-y_1)$ gives us the vertical change. If the line is very steep, then when going from the first point to the second point, we would expect a large vertical change compared to the horizontal change. If the line is not very steep, then when going from the first point to the second point, we would expect a small vertical change compared to the horizontal change.

We are comparing changes. We see that we are comparing

$\begin{array}{lll}\text{The}\text{vertical}\text{change}\hfill & \text{to}\hfill & \text{the}\text{horizontal}\text{change}\hfill \\ \text{The}\text{change}\text{in}y\hfill & \text{to}\hfill & \text{the}\text{change}\text{in}x\hfill \\ y_2-y_1\hfill & \text{to}\hfill & x_2-x_1\hfill \end{array}$