Writing and Graphing Equations in Two Variables

Activity 4 - Linear Equations in Two Variables Part 1

​Solving and graphing linear equations using Standard Form

​Solving and graphing linear equations using functional form!

Activity 4 - Linear Equations in Two Variables Part 1

​Lesson Targets

Students will be able to:

1. Recognize a Solution of an Equation in Two Variables as an Ordered Pair of Values which Make the Equation True

2. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table

3. 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

​Why are some of the equations linear and not linear?

Activity 4 - Linear Equations in Two Variables Part 1

​1. How do we 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 x value or the domain was given and then we used an equation or rule to figure out the y 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.

​What makes this equation a "TRUE" Equation????

Draw

How many different ways can you write the equation:

Activity 4 - Linear Equations in Two Variables Part 1

​1. How do we Recognize a Solution of an Equation in Two Variables as an Ordered Pair of Values which Make the Equation True

- 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.

Let’s start by thinking about the following equation.

​You can re-write this equation as:

​Now we have used the values x and y to show that we have two different values that can be added together to equal y.

Activity 4 - Linear Equations in Two Variables Part 1

​1. How do we Recognize a Solution of an Equation in Two Variables as an Ordered Pair of Values which Make the Equation True​

- Think about ordered pairs. An ordered pair has an x and a y 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 x value is 2 and the y value is 3. The sum is equal to five.

​​

Let’s look at another example...

Let’s start by thinking about the following equation.

​You can re-write this equation as:

​Now we have used the values x and y to show that we have two different values that can be added together to equal y.

Multiple Select

Which of the following would be a set of ordered paired solutions for the following standard form equation:

2x + y = 12

(Hint, there is more than one answer!)

• (2,8)

• (3,6)

• (-5,22)

• (0,6)

Activity 4 - Linear Equations in Two Variables Part 1

​1. How do we Recognize a Solution of an Equation in Two Variables as an Ordered Pair of Values which Make the Equation True​

- Example: Find three solutions to the equation 2x+y=12 and write them in ordered pairs.

​When an equation is written in a form where x and y are added together to equal a third value, we call that standard form. We can say that standard form is Ax+By=C.

Multiple Choice

Which of the following is a formula for the standard form of a straight line?

• y = mx + b

• Ax + By = C

• y - y1 = m(x - x1)

• y = Ax + By

Multiple Choice

Which of the following equations is written in STANDARD form?

• y = 1/2x + 2
• y - 3 = -5(x + 2)
• -7x + 2y = 9
• 3x + 2 = y

Multiple Choice

Solve the following standard form equation substituting in the following ordered pair:

Equation: 2x + y = ?

Ordered Pair: (3,2)

• 12

• 11

• 8

• 9

Multiple Choice

 

Solve the following standard form equation substituting in the following ordered pair:

Equation: $$\frac{1}{2}x\ +\frac{2}{5}y\ =\ ?$$  

Ordered Pair: (4,10)

• 6

• 8

• 10

• 12

Multiple Choice

Solve the following standard form equation substituting in the following ordered pair:

Equation: $$7x\ +\ 3y\ =\ ?$$  

Ordered Pair: (3,2)

• 22

• 24

• 26

• 27

Draw

Draw out the standard form of the equation found below (which is in functional form):

$$y=−4x+6$$  

Remember, change the standard form to: Ax + By = C

Activity 4 - Linear Equations in Two Variables Part 1

2. 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 y value is equal to the rest of the equation.

• To the right is comparing an equation in function form to standard form. We can see that the y value is a function of 2x plus one. This means that the value of y will change based on what the x value is. We can also use f(x) to show that the y value is a function of the rest of the equation. The f(x) is used to substitute for y.

​Standard Form of an Equation

​Functional form of an Equation

Activity 4 - Linear Equations in Two Variables Part 1

2. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table

• ​Exactly, let me try to explain this a little clearer. We know that the value of y depends on the rest of the equation including whichever values we substitute for x. Well, we can say that y is a function of the rest of the equation. Therefore, we can say that the f(x) is also dependent on the rest of the equation. The f(x) is the same as y.

  Let’s look at an example.

​Functional form of an Equation

​How can we find the inputs (domain) and outputs (range) of the following equation?

Draw

Using the following equation, along with using a "T" chart, find the inputs and outputs for the following equation:

$$y=3x+1$$  

Activity 4 - Linear Equations in Two Variables Part 1

2. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table

• Example​

  y=3x+1

  To work with this equation, we have to create a table of values. Then we will know what the value of y is based on the values that we substitute for x.

• Now we have the values for x and y. 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.

​Functional form of an Equation

​Notice how if you substitute the inputs into the equation you will find the output, which makes it a true equation.

Multiple Choice

1. What is the solution for equations below?

y = 3x + 2

y = 5x

• x=2, y=8

• x=5, y=17

• x=1, y=5

• x= -2, y= -4

Activity 4 - Linear Equations in Two Variables Part 1

2. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table

• Sooo... What if the equation is in standard form?

• That is a great question. That means that we will need to rewrite it into function form.

  Let’s look at an example to the right:

Standard Form of an Equation

​How can we rewrite this equation in functional form?

Multiple Choice

Re-write the following equation from standard form to functional form:

$$4x−y=−1$$  

• -1 - 4x = y

• y = 4x + 1

• y = -1x + 4

• y = 4x - 1

Activity 4 - Linear Equations in Two Variables Part 1

2. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table

• 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 4x and the −y to the opposite side of the equals. We can do this by using inverse operations. Remember that an inverse operation is an opposite operation.

Standard Form of an Equation

​How can we rewrite this equation in functional form?

Activity 4 - Linear Equations in Two Variables Part 1

2. Rewrite Equations in Two Variables in Function Form and Find Solutions by Making a Table

• Once we have an equation written in function form, we can use a table of values to figure out a set of ordered pairs.

• ​Now we have a set of ordered pairs for the equation y=4x+1.

Standard Form of an Equation

​How can we rewrite this equation in functional form?

Multiple Choice

Which of the following is an ordered pair for the following functional equation of a straight line?

$$y\ =\ \frac{1}{2}x\ +\ 7$$  

• Ordered Pair (4, 12)

• Ordered Pair (12, 4)

• Ordered Pair (4,9)

• Ordered Pair (9,4)

Multiple Choice

Which of the following is an ordered pair for the following functional equation of a straight line?

$$y\ =\ \frac{2}{5}x\ \ -\ 9$$   

• Ordered Pair (-7,2)

• Ordered Pair (2,-7)

• Ordered Pair (5,-11)

• Ordered Pair (-5,-11)

Multiple Choice

Which of the following T-Charts is the valid T-Chart for the following functional equation:

$$y\ =\ 4x\ -\ 6$$  

Draw

Try and create a graph, along with a T chart and 3 ordered pairs, for the following functional Equation:

$$y\ =\ 4x\ -\ 6$$  

Activity 4 - Linear Equations in Two Variables Part 1

​3. 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.

Activity 4 - Linear Equations in Two Variables Part 1

​3. 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 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.

Activity 4 - Linear Equations in Two Variables Part 1

​3. 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

• 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.

Draw

Graph the line y=2x−1. Tell whether or not this equation is a linear equation.

Activity 4 - Linear Equations in Two Variables Part 1

​3. 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

• 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.

​Ordered Pairs:

Table of Values:

Activity 4 - Linear Equations in Two Variables Part 1

​Ordered Pairs:

Table of Values:

​Linear Equation of a Straight Line:

​Linear Equation of a Straight Line Graphed:

Draw

Create a table of values for each equation and then graph it on the coordinate plane:

$$y\ =\ -2x\ +\ 2$$  

Draw

Create a table of values for each equation and then graph it on the coordinate plane:

$$y\ =\ -\frac{1}{2}x\ +2$$  

Draw

Create a table of values for each equation and then graph it on the coordinate plane:

$$y\ =\ \frac{3}{4}x\ -5$$