Relating Slope and Y-intercept to the Equation of a Line

Let's explore the relationship between the equation of a line and the line's slope and y-intercept.

Consider the linear equation $LaTeX: y=\frac{1}{2}x-1$ . Use the linear equation to complete the table below, and then use the ordered pairs to graph the equation.

$LaTeX: y=\frac{1}{2}x-1$

x	y	(x, y)
–2
0
2
4

To obtain the y-values, we substitute the each x-value into the equation $LaTeX: y=\frac{1}{2}x-1$ .

LaTeX: x= –2: $LaTeX: y=\frac{1}{2}$ (–2) – 1 $LaTeX: \longrightarrow\:y=$ –2

LaTeX: x= 0: $LaTeX: y=\frac{1}{2}$ (0) – 1 $LaTeX: \longrightarrow\:y=$ –1

LaTeX: x= 2: $LaTeX: y=\frac{1}{2}$ (2) – 1 $LaTeX: \longrightarrow\:y=$ 0

LaTeX: x= 4: $LaTeX: y=\frac{1}{2}$ (4) – 1 $LaTeX: \longrightarrow\:y=$ 1

Then we use the calculated y-values to complete the table.

x	y	(x, y)
–2	–2	(–2, –2)
0	–1	(0, –1)
2	0	(2, 0)
4	1	(4, 1)

Next we plot the ordered pairs and connect the points.

$graph with the ordered pairs (-2 -2), (0,-1), (2, 0), and (4, 1) plotted; a line connects the points$

How is the equation of the line related to the slope of the line?

REVIEW: Choose any two points on the line and calculate the slope. Be sure to write the slope as a fraction in reduced form.

ANSWER: Did you get $LaTeX: m=\frac{1}{2}$ for the slope? If not, take a few minutes to review the previous module (Math Interludes VIII).

O.K. the slope of the line is $LaTeX: m=\frac{1}{2}$ . And the equation of line is $LaTeX: y=\frac{1}{2}x-1$ . So we have the following.

$LaTeX: m=\frac{1}{2}$

$LaTeX: y=\frac{1}{2}x-1$

So how is the slope $LaTeX: m=\frac{1}{2}$ related to the equation of the line $LaTeX: y=\frac{1}{2}x-1$ ?

Yes! When LaTeX: y is isolated on one side of the equation, and the other side is simplified, the slope is the number multiplied by LaTeX: x .

How is the equation of the line related to the y-intercept of the line?

If we randomly select and point on the y-axis we see that the x-value is 0. Try, put your finger anywhere on the y-axis. What is the x-value of the point? Starting at the origin and moving toward the point where your finger is resting on the y-axis, we only move up or down. We do not move left or right. So LaTeX: x=0 .

In fact, the y-axis is the only place in the coordinate system where LaTeX: x=0 . Indeed, if , then we know the point is on the y-axis.

Furthermore, the y-intercept occurs where LaTeX: x=0 . So the y-intercept is always the point where the graph of the line crosses the y-axis. Identify the y-intercept on the graph of the line $LaTeX: y=\frac{1}{2}x-1$ .

$graph of the line given by y = (1/2)x-1 with an arrow pointing to the y-intercept (0, -1)$

The graph of the line crosses the x-axis at the point $LaTeX: \left(0,\:-1\right)$ , so the y-intercept is LaTeX: -1 . Since x is always zero at the y-intercept, we use only the y-value to identify the y-intercept.

We have the following.

y-intercept: LaTeX: y=-1

equation: $LaTeX: y=\frac{1}{2}x-1$

So, how is the y-intercept LaTeX: y=-1 related to the equation of the line $LaTeX: y=\frac{1}{2}x-1$ ?

Yes! When LaTeX: y is isolated on one side of the equation, and the other side is simplified, the y-intercept is the number added to the LaTeX: x -term.

Hmmm ... is that word added a little confusing? Afterall, in the equation $LaTeX: y=\frac{1}{2}x-1$ , we subtract 1 from the x-term.

When we look at the equation of the line, $LaTeX: y=\frac{1}{2}x-1$ , we imagine changing the subtraction to add-the-opposite to get $LaTeX: y=\frac{1}{2}x+\left(-1\right)$ . The number we are "adding" to the is LaTeX: x -term is LaTeX: -1 , so the y-intercept is .

The General Equation of a Line (in Slope-intercept Form)

The general equation of a line (in slope-intercept form) is LaTeX: y=mx+b , where LaTeX: m is the slope and LaTeX: b is the y-intercept. For a specific line, is replaced with a number representing the slope, and is replaced with a number representing the y-intercept.

In the equation LaTeX: y=mx+b , LaTeX: x and are the variables. We say that is the dependent variable because its value depends on . And we say that is the independent variable.

On the next page, together we'll learn how to identify the slope and y-intercept for a linear equation.

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