Suppose that you were a senior in high school and wanted to graph a linear equation that could be used to find the number of days until graduation based on the day of the year. One way to graph such an equation would be to find the slope and \begin{align*}y-\end{align*}intercept, but how could you determine this information?

### Slope-Intercept Form

So, you have learned how to graph the solutions to an equation in two variables by making a table and by using its intercepts. A previous lesson introduced the formulas for slope. This lesson will combine intercepts and slope into a new formula.

You have seen different forms of this formula. Below are several examples.

\begin{align*}2x+5& =y\\ y& =\frac{-1}{3} x+11\\ d & =60(h)+45\end{align*}

The proper name given to each of these equations is slope-intercept form because each equation tells the slope and the \begin{align*}y-\end{align*}intercept of the line.

The **slope-intercept form of an equation** is: \begin{align*}y=(slope)x+(y-\end{align*}intercept).

\begin{align*}y=(m)x+b\end{align*}, where \begin{align*}m = slope\end{align*} and \begin{align*}b = y-\end{align*}intercept

This equation makes it quite easy to graph the solutions to an equation of two variables because it gives you two necessary values:

- The starting position of your graph (the \begin{align*}y-\end{align*}intercept)
- The directions to find your second coordinate (the slope)

#### Let's determine the slope and the \begin{align*}y-\end{align*}intercept of the following equations:

- \begin{align*}2x+5=y\end{align*}

Using the definition of slope-intercept form; \begin{align*}2x+5=y\end{align*} has a slope of 2 and a \begin{align*}y-\end{align*}intercept of (0, 5).

- \begin{align*}y=\frac{-1}{3}x+11\end{align*}

\begin{align*}y=\frac{-1}{3}x+11\end{align*} has a slope of \begin{align*}\frac{-1}{3}\end{align*} and a \begin{align*}y-\end{align*}intercept of (0, 11).

- \begin{align*}7x=y\end{align*}.

At first glance, this does not look like the “standard” equation. However, we can substitute values for the slope and \begin{align*}y-\end{align*}intercept.

\begin{align*}7x+0=y\end{align*}

This means the slope is 7 and the \begin{align*}y-\end{align*}intercept is 0.

#### Now, let's determine the slope and \begin{align*}y-\end{align*}intercept of the lines graphed below:

- Line \begin{align*}a\end{align*}: The \begin{align*}y-\end{align*}intercept is (0, 5). The line also passes through (2, 3).

\begin{align*}\text{slope} \ m=\frac{\triangle y}{\triangle x} = \frac{-2}{2}=-1\end{align*}

- Line \begin{align*}b\end{align*}: The \begin{align*}y-\end{align*}intercept is (0, 2). The line also passes through (1, 5).

\begin{align*}\text{slope} \ m = \frac{\triangle y}{\triangle x} = \frac{3}{1}=3\end{align*}

- Line \begin{align*}c\end{align*}: The \begin{align*}y-\end{align*}intercept is (0, -1). The line also passes through (2, 3).

\begin{align*}\text{slope} \ m = \frac{\triangle y}{\triangle x} = \frac{4}{2}=2\end{align*}

- Line \begin{align*}d\end{align*}: The \begin{align*}y-\end{align*}intercept is (0, -3). The line also passes through (4, -4).

\begin{align*}\text{slope} \ m = \frac{\triangle y}{\triangle x} = \frac{-1}{4}=-\frac{1}{4}\end{align*}

### Examples

#### Example 1

Earlier, you were asked how you would determine the slope and \begin{align*}y-\end{align*}intercept of an equation that represents the number of days until graduation based on the day of the year.

To do this, convert the equation into slope-intercept form. If you know how to read an equation in slope-intercept form, then you can easily read what the slope is and what the \begin{align*}y-\end{align*}intercept is. For example, if the equation for this situation was \begin{align*}y=-x + 50\end{align*}, then the slope would be -1 and the \begin{align*}y-\end{align*}intercept would be (0, 50).

#### Example 2

Determine the slope and \begin{align*}y-\end{align*}intercept of \begin{align*}y=5\end{align*}.

Recall that he slope of every line of the form \begin{align*}y=some \ number\end{align*} is zero because it is a horizontal line. Rewriting our original equation to fit slope-intercept form yields:

\begin{align*}y=(0)x+5\end{align*}

Therefore, the slope is zero and the \begin{align*}y-\end{align*}intercept is (0, 5).

You can also use a graph to determine the slope and \begin{align*}y-\end{align*}intercept of a line.

### Review

In 1 – 8, identify the slope and \begin{align*}y-\end{align*}intercept for the equation.

- \begin{align*}y=2x+5\end{align*}
- \begin{align*}y=-0.2x+7\end{align*}
- \begin{align*}y=x\end{align*}
- \begin{align*}y=3.75\end{align*}
- \begin{align*}\frac{2}{3}x-9=y\end{align*}
- \begin{align*}y=-0.01x+10,000\end{align*}
- \begin{align*}7+\frac{3}{5} x=y\end{align*}
- \begin{align*}-5x+12=20\end{align*}

In 9 – 15, identify the slope of the following lines.

- \begin{align*}F\end{align*}
- \begin{align*}C\end{align*}
- \begin{align*}A\end{align*}
- \begin{align*}G\end{align*}
- \begin{align*}B\end{align*}
- \begin{align*}D\end{align*}
- \begin{align*}E\end{align*}

In 16 – 21, identify the slope and \begin{align*}y-\end{align*}intercept for the following lines.

- \begin{align*}D\end{align*}
- \begin{align*}A\end{align*}
- \begin{align*}F\end{align*}
- \begin{align*}B\end{align*}
- \begin{align*}E\end{align*}
- \begin{align*}C\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.9.