9.2: Linear Equations in Two Variables
Introduction
The Omni Theater
While Max and Casey are figuring out the bowling alley information, Tasha and Uniqua are busy coming up with an alternative field trip. They think that the class should go to the Omni Theater instead to see a film on the rainforest. By calling the Science Museum where the theater is located, they discover the following information.
The cost of a ticket is $5.00, but there is a $2.00 service fee per ticket.
“That is a lot of money,” Tasha says.
“Well, it depends on how many students actually go,” Uniqua says.
“Let’s figure it out. There are 22 students in our class and we could have anywhere from 22 to 18 students go based on the number of students absent. Now we need to do the math,” Tasha says taking out a piece of paper.
Here we are working on functions again. To figure out the range of costs with this trip, you will need to write an equation and create a table to show how the cost changes based on the number of students who attend the trip. By the end of this lesson, you will know how to go about solving this problem.
What You Will Learn
By the end of this lesson you will learn how to complete the following skills.
- Recognize a solution of an equation in two variables as an ordered pair of values which make the equation true.
- Rewrite equations in two variables in function form and find solutions by making a table.
- Use a table of solutions to graph equations in two variables, recognizing an equation whose graph is a straight line is called a linear equation, for which each variable only occurs to the first power.
- Recognize equations for graphs of vertical and horizontal lines.
Teaching Time
I. Recognize a Solution of an Equation in Two Variables as an Ordered Pair of Values which Make the Equation True
In the last section, we wrote function rules for function tables. When we did this work, we used a table of values where the \begin{align*}x\end{align*} value or the domain was given and then we used an equation or rule to figure out the \begin{align*}y\end{align*} value or the domain. In this section, we are going to start with equations or rules and then see how these equations can help us to figure out ordered pairs.
Let’s start by thinking about the following equation.
\begin{align*}3+2=5\end{align*}
This is a true equation. You probably remember equations like this one from back in your elementary school days. However, we can look at this equation in a new way. Here we have the statement that three plus two is equal to five. Well, there are other values that could also be added together to equal 5. We could add positive and negative numbers to equal five. Therefore, there are many possible values that could be added together to equal five. Let’s change this equation to one where that is clear.
\begin{align*}x+y=5\end{align*}
Now we have used the values \begin{align*}x\end{align*} and \begin{align*}y\end{align*} to show that we have two different values that can be added together to equal \begin{align*}y\end{align*}.
Think about ordered pairs. An ordered pair has an \begin{align*}x\end{align*} and a \begin{align*}y\end{align*} value. If we were to find values that would make this a true statement, then we could also say that we had ordered pairs that would make this a true statement.
One answer for this equation is the ordered pair (2, 3) where the \begin{align*}x\end{align*} value is 2 and the \begin{align*}y\end{align*} value is 3. The sum is equal to five.
Let’s look at another example.
Example
Find three solutions to the equation \begin{align*}2x+y=12\end{align*} and write them in ordered pairs.
\begin{align*}2 \cdot 2+8=12\end{align*} so the ordered pair is (2, 8).
\begin{align*}2 \cdot 3+6=12\end{align*} so the ordered pair is (3, 6).
\begin{align*}2 \cdot -5+22=12\end{align*} so the ordered pair is (-5, 22).
When an equation is written in a form where \begin{align*}x\end{align*} and \begin{align*}y\end{align*} are added together to equal a third value, we call that standard form. We can say that standard form is \begin{align*}Ax+By=C\end{align*}.
Write the definition for standard form and its equation in your notebook.
II. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table
You just learned how to identify an equation in standard form. We can also write equations in function form. Function form is when the \begin{align*}y\end{align*} value is equal to the rest of the equation.
\begin{align*}y=2x+1\end{align*}
This is an equation in function form. We can see that the \begin{align*}y\end{align*} value is a function of \begin{align*}2x\end{align*} plus one. This means that the value of \begin{align*}y\end{align*} will change based on what the \begin{align*}x\end{align*} value is. We can also use \begin{align*}f(x)\end{align*} to show that the \begin{align*}y\end{align*} value is a function of the rest of the equation. The \begin{align*}f(x)\end{align*} is used to substitute for \begin{align*}y\end{align*}.
Exactly, let me try to explain this a little clearer. We know that the value of \begin{align*}y\end{align*} depends on the rest of the equation including whichever values we substitute for \begin{align*}x\end{align*}. Well, we can say that \begin{align*}y\end{align*} is a function of the rest of the equation. Therefore, we can say that the \begin{align*}f(x)\end{align*} is also dependent on the rest of the equation. The \begin{align*}f(x)\end{align*} is the same as \begin{align*}y\end{align*}.
Let’s look at an example.
Example
\begin{align*}y=3x+1\end{align*}
To work with this equation, we have to create a table of values. Then we will know what the value of \begin{align*}y\end{align*} is based on the values that we substitute for \begin{align*}x\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 4 |
2 | 7 |
3 | 10 |
4 | 13 |
5 | 16 |
Now we have the values for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. You can also notice that since we have these two values, we also have a set of ordered pairs that have been created in a table form.
That is a great question. That means that we will need to rewrite it into function form.
Let’s look at an example.
Example
\begin{align*}4x-y=-1\end{align*}
Here we have an equation in standard form. We will need to rewrite this equation into function form. To do this, we will move the negative one with the \begin{align*}4x\end{align*} and the \begin{align*}-y\end{align*} to the opposite side of the equals. We can do this by using inverse operations. Remember that an inverse operation is an opposite operation.
\begin{align*}4x-y+y &=-1+y \\ 4x &=-1+y \\ 4x+1 &=y \ or \ y=4x+1\end{align*}
Once we have an equation written in function form, we can use a table of values to figure out a set of ordered pairs.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 1 |
1 | 5 |
2 | 9 |
3 | 13 |
Now we have a set of ordered pairs for the equation \begin{align*}y=4x+1\end{align*}.
III. Use a Table of Solutions to Graph Equations in Two Variables, Recognizing an Equation whose Graph is a Straight Line is Called a Linear Equation, for which Each Variable Only Occurs to the First Power
Now we have looked at equations in standard form. We have rewritten these equations into function form. Then we have created a table of values for an equation which makes this equation true. Our next step is to graph the ordered pairs from a table of values. We graph these values on the coordinate plane. Let’s look at the last example.
Example
\begin{align*}y=4x+1\end{align*}
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | 1 |
1 | 5 |
2 | 9 |
3 | 13 |
Now we have an equation in function form. We have a table of values that makes this equation true, so we can look at graphing this function.
First, let’s write out the ordered pairs from the table. Notice that these values are all positive, however, you can have positive and negative values that make an equation true.
(0, 1)
(1, 5)
(2, 9)
(3, 13)
Notice that the graph of this equation forms a straight line. When a set of values are graphed to represent an equation, if a straight line is created, we call this a linear equation. Linear means line.
Let’s look at another example.
Example
Graph the line \begin{align*}y=2x-1\end{align*}. Tell whether or not this equation is a linear equation.
First, notice that this equation is already in function form, so we can create a table of values. This table will give us our ordered pairs for graphing.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
0 | -1 |
1 | 1 |
2 | 3 |
3 | 5 |
Here are the ordered pairs.
(0, -1)
(1, 1)
(2, 3)
(3, 5)
Now let’s graph the equation.
You can see from the graph that this is a linear equation. The graph of the line is a straight line.
IV. Recognize Equations for Graphs of Vertical and Horizontal Lines
Now you know how to graph a linear equation and how to identify this equation based on the graph. What about an equation like \begin{align*}y=5\end{align*} or \begin{align*}x=-3?\end{align*} These are equations we don’t need to make tables. If \begin{align*}y\end{align*} is equal to five, then we could say that \begin{align*}x\end{align*} is equal to all of the other values.
Now let’s look at the graph of this equation.
You can see that the graph of this line is a horizontal line.
Next, we can look at the equation \begin{align*}x= -3\end{align*}. Now we can graph this line. If we know that the \begin{align*}x\end{align*} value is -3, then we can say that all of the other \begin{align*}y\end{align*} values are all of the other numbers. Let’s take a look at this graph.
The graph of the equation \begin{align*}x = -3\end{align*} is a vertical line.
We can say that when \begin{align*}x\end{align*} is equal to one value, then we have a vertical line and when \begin{align*}y\end{align*} is equal to one value that we have a horizontal line.
Write this information down in your notebook. Be sure that you understand how to identify the graph of a horizontal or a vertical line.
Real-Life Example Completed
The Omni Theater
Here is the original problem once again. Reread it and then write an equation and a table of values to show the varying costs.
While Max and Casey are figuring out the bowling alley information, Tasha and Uniqua are busy coming up with an alternative field trip. They think that the class should go to the Omni Theater instead to see a film on the rainforest. By calling the Science Museum where the theater is located, they discover the following information.
The cost of a ticket is $5.00, but there is a $2.00 service fee per ticket.
“That is a lot of money,” Tasha says.
“Well, it depends on how many students actually go,” Uniqua says.
“Let’s figure it out. There are 22 students in our class and we could have anywhere from 22 to 18 students go based on the number of students absent. Now we need to do the math,” Tasha says taking out a piece of paper.
Remember, there are two parts to your problem.
Solution to Real – Life Example
The problem can first be solved by writing an equation. To do this, we need to look at the given information.
Total cost \begin{align*}= y\end{align*}
\begin{align*}x = \end{align*} number of students and this can vary
$2.00 is the service fee
$5.00 per ticket
\begin{align*}y=5x+2\end{align*}
Now we can create our table of values based on the range of students in attendance. Uniqua and Tasha think that there will be anywhere from 22 to 18 students on the trip.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
22 | $112 |
21 | $107 |
20 | $102 |
19 | $97 |
18 | $92 |
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Ordered pair
- the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values that can be found in a table or used to graph points or a line on the coordinate plane.
- Standard form
- the form of an equation \begin{align*}Ax+By=C\end{align*}
- Function form
- the form of an equation \begin{align*}y=mx+b\end{align*}
- Linear Equation
- the graph of a linear equation forms a straight line.
Time to Practice
Directions: Find 4 solutions to the function \begin{align*}3x+y=24\end{align*}. Write your answers as ordered pairs.
Directions: Find 4 solutions to the function \begin{align*}2x-y=9\end{align*}. Write your answers are ordered pairs.
Directions: Write each equation in standard form.
- \begin{align*}y=2x+3\end{align*}
- \begin{align*}y=-4x+6\end{align*}
- \begin{align*}y=-2x-4\end{align*}
- \begin{align*}y=-5x+4\end{align*}
- \begin{align*}y=-3x-2\end{align*}
- \begin{align*}y=-4x+6\end{align*}
Directions: Create a table of values for each equation and then graph it on the coordinate plane.
- \begin{align*}y=2x+1\end{align*}
- \begin{align*}y=3x+1\end{align*}
- \begin{align*}y=4x\end{align*}
- \begin{align*}y= -2x+2\end{align*}
- \begin{align*}y=2x-2\end{align*}
- \begin{align*}y=x-1\end{align*}
- \begin{align*}x=4\end{align*}
- \begin{align*}y= -2\end{align*}
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