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# 9.5: Slope-Intercept Form

Difficulty Level: At Grade Created by: CK-12

## Introduction

Scientists in the Rainforest

The rainforest was such a popular topic of discussion that Mr. Thomas let the students talk about it all week. They loved discussing all of the things that they had seen. One day, they began talking about the scientists and all of the things that they had to carry with them into the rainforest.

“You know, they couldn’t exactly run out to the store to pick something up,” Casey commented.

“Or call for a pizza!” Susan said.

Mr. Thomas once again seized a great opportunity to write the following problem on the board.

A group of backpackers leaves with 84lbs. of food. They plan to eat 11lbs. per day. Use an equation to show on a graph how much food they should have after each day. How long should their food last?

To work on this problem, you will need to know about writing equations involving slope. The food changes per day based on how much the backpackers eat. This lesson will teach you all that you need to know to work on this problem.

What You Will Learn

By the end of this lesson, you will be able to complete the following skills.

• Use a graph to find the slope and y\begin{align*}y \end{align*} – intercept of a linear equation given in function form.
• Calculate the slopes and y\begin{align*}y \end{align*} – intercepts of linear equations in a variety of forms, and recognize slope – intercept form as equivalent to function form.
• Graph linear equations given in a variety of forms using slope and y\begin{align*}y\end{align*} – intercept.
• Solve real – world problems using slope – intercept forms of linear equations.

Teaching Time

I. Use a Graph to Find the Slope and y\begin{align*}\underline{y}\end{align*} – Intercept of a Linear Equation Given in Function Form

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 covering in this lesson.

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.

Let’s look at an example of the slope – intercept form in action.

Example

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.

II. Calculate the Slopes and y\begin{align*}\underline{y}\end{align*} – Intercepts of Linear Equations in a Variety of Forms, Recognize Slope – Intercept Form as Equivalent to Function Form

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.

Example

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\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 y\begin{align*}y\end{align*} – intercept from the equation.

23=slope=yintercept\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 =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!

III. Graph Linear Equations Given in a Variety of Forms Using Slope and y\begin{align*}\underline{y}\end{align*} – Intercept

Do you see how useful the slope-intercept form, y=mx+b\begin{align*}y=mx+b\end{align*}, is to find the slope and y\begin{align*}y\end{align*}-intercept? Using this form, graphing is going to be easy, too. Since we know the slope and we know the y\begin{align*}y\end{align*}-intercept, then instead of using a table of values, we can plot the y\begin{align*}y\end{align*}-intercept on the coordinate plane and find our next point using the slope.

Example

Graph the line that goes with the equation y=x+5\begin{align*}y=-x+5\end{align*}

First, we can determine that the slope is -1 and the y\begin{align*}y \end{align*} – intercept is 5. Next, we can graph this line using these two pieces of information.

We can also graph lines in a different form. First, we will need to rewrite them into slope – intercept form. Then we can graph the equation.

Example

Graph the line 3x+y=9\begin{align*}3x+y=9\end{align*}

First, we rewrite this equation from standard form to slope – intercept form. We do this by using inverse operations.

3x3x+yy=3x+9=3x+9\begin{align*}3x-3x+y &=-3x+9\\ y &= -3x+9\end{align*}

Now we know that the slope is -3 and the y\begin{align*}y \end{align*} – intercept is 9. Next, we can graph the equation of the line.

Now we can look at how linear equations can be found in real – life situations.

IV. Solve Real – World Problems Using Slope – Intercept Forms of Linear Equations

The slope-intercept form allows can be useful in solving many problems in the real world. Let’s look at an example

Example

A store employee gets paid $12.50 per hour plus a weekly bonus of$50 for punctuality. Assuming the employee is on time every day, graph her wages earned. How much would she earn for working 10 hours? 20 hours? 30 hours?

We could use the equation w=12.5h+50\begin{align*}w=12.5h+50\end{align*}.

The graph shows that wages for 10 hours would be $175, for 20 hours$300 and for 30 hours 425. ## Real-Life Example Completed Scientists in the Rainforest Here is the original problem once again. Reread it and then solve Mr. Thomas’ problem using what you have learned in this lesson. The rainforest was such a popular topic of discussion that Mr. Thomas let the students talk about it all week. They loved discussing all of the things that they had seen. One day, they began talking about the scientists and all of the things that they had to carry with them into the rainforest. “You know, they couldn’t exactly run out to the store to pick something up,” Casey commented. “Or call for a pizza!” Susan said. Mr. Thomas once again seized a great opportunity to write the following problem on the board. A group of backpackers leaves with 84lbs. of food. They plan to eat 11lbs. per day. Use an equation to show on a graph how much food they should have after each day. How long should their food last? Solution to Real – Life Example First, we need to write an equation to represent the food. Use the equation f=11d+84\begin{align*}f=-11d+84\end{align*} where f\begin{align*}f\end{align*} is the food they have remaining and d\begin{align*}d\end{align*} is the number of days that have been gone. The graph shows that they have enough food for a little more than seven days. ## Vocabulary Here are the vocabulary words that are found in this lesson. 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. ## Time to 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. y=5x1\begin{align*}y=5x-1\end{align*} 5. y=12x+2\begin{align*}y=\frac{1}{2}x+2\end{align*} 6. y=2x+4\begin{align*}y= -2x+4\end{align*} 7. y=3x1\begin{align*}y= -3x-1\end{align*} 8. y=13x+5\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. 2x+4y=12\begin{align*}2x+4y=12\end{align*} 1. Write this equation in slope – intercept form. 2. What is the slope? 3. What is the y\begin{align*}y\end{align*} – intercept? 2x+2y=8\begin{align*}2x+2y=8\end{align*} 1. Write this equation in slope – intercept form. 2. What is the slope? 3. What is the y\begin{align*}y\end{align*} – intercept? 3x+6y=2\begin{align*}3x+6y=2\end{align*} 1. Write the following equation in slope – intercept form. 2. What is the slope? 3. What is the y\begin{align*}y\end{align*} – intercept? Directions: Use what you have learned to solve each problem. Miguel wants to save47 for a video game. He received $20 as a gift and gets$4 per week for allowance.

1. Write an equation in slope-intercept form that represents this situation.
2. Graph the equation.
3. How long will it take him to save enough money?

A homeowner wants to reduce the amount of electricity he uses at his house. He finds that he uses 600 watts of power per month. By using energy-efficient light bulbs, he decreases his usage by 12 watts per light bulb each month.

1. How many energy-efficient light bulbs does he need to cut his consumption in half?
2. Write an equation in slope-intercept form.

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