Do you know how to identify the slope in an equation? How about the y-intercept?

Take a look at this situation.

\begin{align*}y=-2x-8\end{align*}

**In this Concept, you will learn how to figure out the slope and the y-intercept by looking at an equation. We will look at this one again at the end of the Concept.**

### Guidance

We have seen linear equations in function form, have created tables of values and graphs to represent them, looked at their \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts, and studied their slopes. One of the most useful forms of a linear equation is the ** slope-intercept form** which we will be using with standard form in this Concept.

Remember ** standard form**?

**The standard form of an equation is when the equation is written in \begin{align*}Ax+By=C\end{align*} form.**

This form of the equation allows us to find many possible solutions. In essence, we could substitute any number of values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} and create the value for \begin{align*}C\end{align*}. When an equation is written in standard form, it is challenging for us to determine the slope and the \begin{align*}y\end{align*} – intercept.

**Think back, remember that the** *slope***is the steepness of the line and the** *\begin{align*}y\end{align*} – intercept***is the point where the line crosses the \begin{align*}y\end{align*} – axis.**

**We can write an equation in a different form than in standard form. This is when \begin{align*}y = \end{align*} an equation. We call this form of an equation** *slope – intercept form***.**

**Slope – Intercept Form is \begin{align*}y=mx+b\end{align*} – where \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y\end{align*} – intercept.**

Take a look at this graph and equation.

Graph the line \begin{align*}y=3x+1\end{align*}

**Here we can calculate the slope of the line using the rise over the run and see that it is 3. The \begin{align*}y\end{align*} – intercept is 1. Notice that we can find these values in our equation too.**

*When an equation is in slope – intercept form, we can spot the slope and the \begin{align*}y \end{align*} – intercept by looking at the equation.*

\begin{align*}y={\color{red}m}x+ {\color{cyan}b}\end{align*}

Here \begin{align*}m\end{align*} is the value of the slope and \begin{align*}b\end{align*} is the value of the \begin{align*}y\end{align*} – intercept.

For any equation written in the form \begin{align*}y=mx+b\end{align*}, \begin{align*}m\end{align*} is the slope and \begin{align*}b\end{align*} is the \begin{align*}y\end{align*}-intercept. For that reason, \begin{align*}y=mx+b\end{align*} is called the *slope-intercept form.* Using the properties of equations, you can write any equation in this form.

**Because we can use slope – intercept form, we can rewrite equations in standard form into slope – intercept form. Then we can easily determine the slope and \begin{align*}y\end{align*} – intercept of each equation.**

Take a look here.

Write \begin{align*}4x+2y=6\end{align*} in slope – intercept form. Then determine the slope and the \begin{align*}y\end{align*} – intercept by using the equation.

\begin{align*}4x+2y &=6 \\ 4x+2y-2y &=6-2y\\ 4x &=6-2y\\ 4x-6 &=-2y\\ \frac{4x-6}{-2} &=y \\ y &= -2x+3\end{align*}

**Now we can determine the slope and the \begin{align*}y\end{align*} – intercept from the equation.**

\begin{align*}-2 &= \text{slope}\\ 3 &= y - \text{intercept}\end{align*}

**Think back to our work with functions. Remember how we could write a function in function form? Well take a look at function form compared with slope – intercept form.**

**Function form \begin{align*}= f(x)=2x+1\end{align*}**

**Slope – Intercept Form \begin{align*}= y=2x+1\end{align*}**

**Yes! The two are the same. These two equations are equivalent!**

Determine the slope and the y-intercept in each equation.

#### Example A

\begin{align*}y=x+4\end{align*}

**Solution: slope = 1, y-intercept = 4**

#### Example B

\begin{align*}2x+y=10\end{align*}

**Solution: slope = -2, y-intercept = 10**

#### Example C

\begin{align*}-3x+y=9\end{align*}

**Solution: slope = 3, y-intercept = 9**

Now let's go back to the dilemma at the beginning of the Concept.

\begin{align*}y=-2x-8\end{align*}

**Looking at this equation, you can see that the slope is \begin{align*}-2\end{align*} and the y-intercept is \begin{align*}8\end{align*}.**

### Vocabulary

- Slope – Intercept Form
- the form of an equation \begin{align*}y=mx+b\end{align*}

- Standard Form
- the form of an equation \begin{align*}Ax+By=C\end{align*}

- Slope
- the steepness of the line, calculated by the ratio of rise over run.

- \begin{align*}y \end{align*} – Intercept
- the point where a line crosses the \begin{align*}y\end{align*}axis.

### Guided Practice

Here is one for you to try on your own.

Write this equation in slope-intercept form and then determine the slope and the y-intercept.

\begin{align*}-18x+6y &= 12 \\ -18x+6y &=12 \\ -18x+6y+18x &=18x+12\\ 6y &=18x+12\\ 6y&=18x+12\\ \frac{18x+12}{6} &=y \\ y &= 3x+2\end{align*}

**Given this equation, the slope is 3 and the y-intercept is 2.**

### Video Review

Converting to Slope-Intercept Form

### Practice

Directions: Look at each equation and identify the slope and the \begin{align*}y \end{align*} – intercept by looking at each equation. There are two answers for each problem.

- \begin{align*}y=2x+4\end{align*}
- \begin{align*}y=3x-2\end{align*}
- \begin{align*}y=4x+3\end{align*}
- \begin{align*}y=5x-1\end{align*}
- \begin{align*}y=\frac{1}{2}x+2\end{align*}
- \begin{align*}y= -2x+4\end{align*}
- \begin{align*}y= -3x-1\end{align*}
- \begin{align*}y=\frac{-1}{3}x+5\end{align*}

Directions: Use what you have learned to write each in slope – intercept form and then answer each question.

- \begin{align*}2x+4y=12\end{align*}
- Write this equation in slope – intercept form.
- What is the slope?
- What is the \begin{align*}y\end{align*} – intercept?
- \begin{align*}6x+3y=24\end{align*}
- Write this equation in slope – intercept form.
- What is the slope?
- What is the \begin{align*}y\end{align*} – intercept?
- \begin{align*}5x+5y=15\end{align*}
- Write this equation in slope – intercept form.
- What is the slope?
- What is the \begin{align*}y\end{align*} – intercept?