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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.

\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.

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

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.

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 \begin{align*}y \end{align*} – intercept by looking at each equation. There are two answers for each problem.

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

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