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# Slope-Intercept Form

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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.

$y=-2x-8$

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 $x$ - and $y$ -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 $Ax+By=C$ form.

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

Think back, remember that the slope is the steepness of the line and the $y$ – intercept is the point where the line crosses the $y$ – axis.

We can write an equation in a different form than in standard form. This is when $y =$ an equation. We call this form of an equation slope – intercept form .

Slope – Intercept Form is $y=mx+b$ – where $m$ is the slope and $b$ is the $y$ – intercept.

Take a look at this graph and equation.

Graph the line $y=3x+1$

Here we can calculate the slope of the line using the rise over the run and see that it is 3. The $y$ – 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 $y$ – intercept by looking at the equation.

$y={\color{red}m}x+ {\color{cyan}b}$

Here $m$ is the value of the slope and $b$ is the value of the $y$ – intercept.

For any equation written in the form $y=mx+b$ , $m$ is the slope and $b$ is the $y$ -intercept. For that reason, $y=mx+b$ 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 $y$ – intercept of each equation.

Take a look here.

Write $4x+2y=6$ in slope – intercept form. Then determine the slope and the $y$ – intercept by using the equation.

$4x+2y &=6 \\4x+2y-2y &=6-2y\\4x &=6-2y\\4x-6 &=-2y\\\frac{4x-6}{-2} &=y \\y &= -2x+3$

Now we can determine the slope and the $y$ – intercept from the equation.

$-2 &= \text{slope}\\3 &= y - \text{intercept}$

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 $= f(x)=2x+1$

Slope – Intercept Form $= y=2x+1$

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

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

#### Example A

$y=x+4$

Solution: slope = 1, y-intercept = 4

#### Example B

$2x+y=10$

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

#### Example C

$-3x+y=9$

Solution: slope = 3, y-intercept = 9

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

$y=-2x-8$

Looking at this equation, you can see that the slope is $-2$ and the y-intercept is $8$ .

### Vocabulary

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

$-18x+6y &= 12 \\-18x+6y &=12 \\-18x+6y+18x &=18x+12\\6y &=18x+12\\6y&=18x+12\\\frac{18x+12}{6} &=y \\y &= 3x+2$

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 $y$ – intercept by looking at each equation. There are two answers for each problem.

1. $y=2x+4$
2. $y=3x-2$
3. $y=4x+3$
4. $y=5x-1$
5. $y=\frac{1}{2}x+2$
6. $y= -2x+4$
7. $y= -3x-1$
8. $y=\frac{-1}{3}x+5$

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

1. $2x+4y=12$
2. Write this equation in slope – intercept form.
3. What is the slope?
4. What is the $y$ – intercept?
5. $6x+3y=24$
6. Write this equation in slope – intercept form.
7. What is the slope?
8. What is the $y$ – intercept?
9. $5x+5y=15$
10. Write this equation in slope – intercept form.
11. What is the slope?
12. What is the $y$ – intercept?