Lesson Objectives

- Demonstrate an understanding of how to solve a linear equation in one variable
- Learn the definition of a linear equation in two variables
- Learn how to write the solution for a linear equation in two variables using an ordered pair (x,y)
- Learn how to check a solution for a linear equation in two variables
- Learn how to create a table of ordered pairs for a linear equation in two variables

## Linear Equation in Two Variables Definition

Up to this point, we have exclusively worked with equations that contained one variable only. When we solved these equations, we normally ended up with a single solution. Now we will begin to work with more complex equations. A linear equation in two variables is an equation such as:

3x + 5y = 21

Notice how we now have two variables involved (x and y). We have a special method of writing the solution for a linear equation in two variables. This format is known as an ordered pair. As an example, one solution to our equation is:

x = 2, y = 3

This can be written as an ordered pair as:

(2,3)

An ordered pair is listed as the x value first or on the left, and the y value last or on the right. To check if (2,3) is a solution for this equation, we plug in a 2 for x and a 3 for y and simplify. Just as before, we are looking for the left and right sides to be the same value.

3x + 5y = 21

(2,3)

3(2) + 5(3) = 21

6 + 15 = 21

21 = 21

Since we have the same value on each side, we can say that (2,3) is a solution for our equation. When we have a linear equation in two variables, there are an infinite number of solutions. This means there are an unlimited number of (x,y) ordered pairs that will satisfy the equation. As another example, we can show that (7,0) also works as a solution.

3(7) + 5(0) = 21

21 + 0 = 21

21 = 21

Let's take a look at a few examples.

Example 1: Determine if each ordered pair is a solution for the equation.

-4x + y = 10

(-3,2)

Let's plug in a (-3) for x and a 2 for y:

-4(-3) + 2 = 10

12 + 2 = 10

14 = 10 (false)

(-3,2) is not a solution.

(-2,2)

Let's plug in a (-2) for x and a 2 for y:

-4(-2) + 2 = 10

8 + 2 = 10

10 = 10

(-2,2) is a solution.

(0,10)

Let's plug in a 0 for x and a 10 for y:

-4(0) + 10 = 10

10 = 10

(0,10) is a solution.

Example 2: Determine if each ordered pair is a solution for the equation.

3x - 12y = -3

(3,1)

Let's plug in a (3) for x and a 1 for y:

3(3) - 12(1) = -3

9 - 12 = -3

-3 = -3 (true)

(3,1) is a solution.

(4, -2)

Let's plug in a 4 for x and a (-2) for y:

3(4) - 12(-2) = -3

12 + 24 = -3

36 = -3 (false)

(4,-2) is not a solution.

(1,0)

Let's plug in a 1 for x and a 0 for y:

3(1) - 12(0) = -3

3 - 0 = -3

3 = -3 (false)

(1,0) is not a solution.

2x - 5y = 20

Let's suppose we want to create three ordered pair solutions for our equation. What can we do here? We can choose a value for x and solve for y or we can choose a value for y and solve for x. It is usually easy to use 0 when picking a value. Let's see if we can fill in the blank for this ordered pair:

(0, __)

Plug in a 0 for x and solve for y:

2(0) - 5y = 20

0 - 5y = 20

-5y = 20

-5/-5 y = 20/-5

y = -4

(0,-4) is a solution.

Now let's try to use 0 as a value for y and solve for the unknown x. Let's fill in the blank for this ordered pair:

(__ ,0)

2x - 5(0) = 20

2x = 20

2/2 x = 20/2

x = 10

(10, 0) is a solution.

Let's try one more, and get a third and final ordered pair solution. Let's let y = 2. Let's fill in the blank for this ordered pair:

(__ ,2)

2x - 5(2) = 20

2x - 10 = 20

2x = 30

2/2 x = 30/2

x = 15

(15, 2) is a solution.

Let's look at an example.

Example 3: Complete each table of values.

-3x + y = -4

Let's start with (0,_):

-3(0) + y = -4

0 + y = -4

y = -4

(0,-4)

Let's move to (_,-1):

-3x + (-1) = -4

-3x = -3

-3/-3 x = -3/-3

x = 1

(1,-1)

Let's end with (5,_):

-3(5) + y = -4

-15 + y = -4

y = 11

(5,11)

Let's fill in our table:

3x + 5y = 21

Notice how we now have two variables involved (x and y). We have a special method of writing the solution for a linear equation in two variables. This format is known as an ordered pair. As an example, one solution to our equation is:

x = 2, y = 3

This can be written as an ordered pair as:

(2,3)

An ordered pair is listed as the x value first or on the left, and the y value last or on the right. To check if (2,3) is a solution for this equation, we plug in a 2 for x and a 3 for y and simplify. Just as before, we are looking for the left and right sides to be the same value.

3x + 5y = 21

(2,3)

3(2) + 5(3) = 21

6 + 15 = 21

21 = 21

Since we have the same value on each side, we can say that (2,3) is a solution for our equation. When we have a linear equation in two variables, there are an infinite number of solutions. This means there are an unlimited number of (x,y) ordered pairs that will satisfy the equation. As another example, we can show that (7,0) also works as a solution.

3(7) + 5(0) = 21

21 + 0 = 21

21 = 21

Let's take a look at a few examples.

Example 1: Determine if each ordered pair is a solution for the equation.

-4x + y = 10

(-3,2)

Let's plug in a (-3) for x and a 2 for y:

-4(-3) + 2 = 10

12 + 2 = 10

14 = 10 (false)

(-3,2) is not a solution.

(-2,2)

Let's plug in a (-2) for x and a 2 for y:

-4(-2) + 2 = 10

8 + 2 = 10

10 = 10

(-2,2) is a solution.

(0,10)

Let's plug in a 0 for x and a 10 for y:

-4(0) + 10 = 10

10 = 10

(0,10) is a solution.

Example 2: Determine if each ordered pair is a solution for the equation.

3x - 12y = -3

(3,1)

Let's plug in a (3) for x and a 1 for y:

3(3) - 12(1) = -3

9 - 12 = -3

-3 = -3 (true)

(3,1) is a solution.

(4, -2)

Let's plug in a 4 for x and a (-2) for y:

3(4) - 12(-2) = -3

12 + 24 = -3

36 = -3 (false)

(4,-2) is not a solution.

(1,0)

Let's plug in a 1 for x and a 0 for y:

3(1) - 12(0) = -3

3 - 0 = -3

3 = -3 (false)

(1,0) is not a solution.

### Creating a Table of Ordered Pairs

Normally we are not given solutions for our equations, which means we have to generate them. In a few lessons, we will begin graphing linear equations in two variables. In order to be successful in that lesson, we need to understand how to create a table of ordered pairs that satisfy a given linear equation in two variables. Let's suppose we have the following equation:2x - 5y = 20

Let's suppose we want to create three ordered pair solutions for our equation. What can we do here? We can choose a value for x and solve for y or we can choose a value for y and solve for x. It is usually easy to use 0 when picking a value. Let's see if we can fill in the blank for this ordered pair:

(0, __)

Plug in a 0 for x and solve for y:

2(0) - 5y = 20

0 - 5y = 20

-5y = 20

-5/-5 y = 20/-5

y = -4

(0,-4) is a solution.

Now let's try to use 0 as a value for y and solve for the unknown x. Let's fill in the blank for this ordered pair:

(__ ,0)

2x - 5(0) = 20

2x = 20

2/2 x = 20/2

x = 10

(10, 0) is a solution.

Let's try one more, and get a third and final ordered pair solution. Let's let y = 2. Let's fill in the blank for this ordered pair:

(__ ,2)

2x - 5(2) = 20

2x - 10 = 20

2x = 30

2/2 x = 30/2

x = 15

(15, 2) is a solution.

Let's look at an example.

Example 3: Complete each table of values.

-3x + y = -4

x | y |
---|---|

0 | _ |

_ | -1 |

5 | _ |

-3(0) + y = -4

0 + y = -4

y = -4

(0,-4)

Let's move to (_,-1):

-3x + (-1) = -4

-3x = -3

-3/-3 x = -3/-3

x = 1

(1,-1)

Let's end with (5,_):

-3(5) + y = -4

-15 + y = -4

y = 11

(5,11)

Let's fill in our table:

x | y |
---|---|

0 | -4 |

1 | -1 |

5 | 11 |

#### Skills Check:

Example #1

Which ordered pair is a solution to the given equation. $$5x - 2y=40$$

Please choose the best answer.

A

$$\left(1, -\frac{1}{2}\right)$$

B

$$\left(5, -\frac{15}{2}\right)$$

C

$$\left(-2, 7\right)$$

D

$$\left(20, 0\right)$$

E

$$\left(8, -8\right)$$

Example #2

Which ordered pair is a solution to the given equation. $$-4x + 3y=16$$

Please choose the best answer.

A

$$\left(0, -\frac{4}{3}\right)$$

B

$$\left(\frac{2}{3}, \frac{56}{9}\right)$$

C

$$\left(-1, 5\right)$$

D

$$\left(\frac{3}{16}, 0\right)$$

E

$$\left(-4, \frac{16}{3}\right)$$

Example #3

Which ordered pair is a solution to the given equation. $$12x - 3y=1$$

Please choose the best answer.

A

$$\left(0, -5\right)$$

B

$$\left(-\frac{1}{3}, 4\right)$$

C

$$\left(\frac{1}{4}, -1\right)$$

D

$$\left(-1, -\frac{13}{3}\right)$$

E

$$\left(\frac{1}{3}, 0\right)$$

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