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# Slope-Intercept Form

## y = mx +b. Finding slope(m) and y-intercept(b) from equation or graph

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Practice Slope-Intercept Form
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Using Slope-Intercept Form

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

Take a look at this situation.

y=2x8\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 x\begin{align*}x\end{align*}- and y\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 Ax+By=C\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 x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} and create the value for C\begin{align*}C\end{align*}. When an equation is written in standard form, it is challenging for us to determine the slope and the y\begin{align*}y\end{align*} – intercept.

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

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

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

Take a look at this graph and equation.

Graph the line y=3x+1\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 y\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 y\begin{align*}y \end{align*} – intercept by looking at the equation.

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

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

For any equation written in the form y=mx+b\begin{align*}y=mx+b\end{align*}, m\begin{align*}m\end{align*} is the slope and b\begin{align*}b\end{align*} is the y\begin{align*}y\end{align*}-intercept. For that reason, y=mx+b\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 y\begin{align*}y\end{align*} – intercept of each equation.

Take a look here.

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

4x+2y4x+2y2y4x4x64x62y=6=62y=62y=2y=y=2x+3

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

23=slope=yintercept

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

Slope – Intercept Form =y=2x+1\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

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

Solution: slope = 1, y-intercept = 4

#### Example B

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

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

#### Example C

3x+y=9\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.

y=2x8\begin{align*}y=-2x-8\end{align*}

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

### Vocabulary

Slope – Intercept Form
the form of an equation y=mx+b\begin{align*}y=mx+b\end{align*}
Standard Form
the form of an equation Ax+By=C\begin{align*}Ax+By=C\end{align*}
Slope
the steepness of the line, calculated by the ratio of rise over run.
y\begin{align*}y \end{align*} – Intercept
the point where a line crosses the y\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.

18x+6y18x+6y18x+6y+18x6y6y18x+126y=12=12=18x+12=18x+12=18x+12=y=3x+2

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

### Practice

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

1. y=2x+4\begin{align*}y=2x+4\end{align*}
2. y=3x2\begin{align*}y=3x-2\end{align*}
3. y=4x+3\begin{align*}y=4x+3\end{align*}
4. \begin{align*}y=5x-1\end{align*}
5. \begin{align*}y=\frac{1}{2}x+2\end{align*}
6. \begin{align*}y= -2x+4\end{align*}
7. \begin{align*}y= -3x-1\end{align*}
8. \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.

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

### Vocabulary Language: English

$y-$intercept

$y-$intercept

A $y-$intercept is a location where a graph crosses the $y-$axis. As a coordinate pair, this point will always have the form $(0, y)$.
Slope

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$
Slope-Intercept Form

Slope-Intercept Form

The slope-intercept form of a line is $y = mx + b,$ where $m$ is the slope and $b$ is the $y-$intercept.
Standard Form

Standard Form

The standard form of a line is $Ax + By = C$, where $A, B,$ and $C$ are real numbers.