Lesson Objectives

- Demonstrate an understanding of how to create a table of Ordered Pairs for a Linear Equation in Two Variables
- Demonstrate an understanding of how to Plot an Ordered Pair
- Learn how to graph a Linear Equation in Two Variables
- Learn how to find the x-intercept and y-intercept
- Learn how to graph horizontal lines and vertical lines

## How to Graph a Linear Equation in Two Variables

Over the last few lessons, we have learned about linear equations in two variables. First and foremost,
we learned this new type of equation had two variables, x, and y. As an example:

-2x - y = 4

We write a solution for a linear equation in two variables as an ordered pair (x,y). When we encounter a linear equation in two variables, there is an infinite number of solutions. To obtain a solution, we can choose a value for x and solve for the unknown y. Let's suppose we take our above equation and choose a value of 1 for x. This gives us the ordered pair:

(1, __)

The blank is the unknown y value. We can just plug in a 1 for x in our equation and simplify:

-2(1) - y = 4

-2 - y = 4

-y = 4 + 2

-y = 6

-1/-1 y = 6/-1

y = -6

This tells us our unknown y value is (-6). This means our ordered pair is:

(1,-6)

We can check by plugging in a 1 for x and a (-6) for y:

-2(1) - (-6) = 4

-2 + 6 = 4

4 = 4

Since we can make as many ordered pair solutions for a linear equation in two variables as we would like, it is normal to graph the equation. This allows us to get a visual representation of all solutions for the equation. To graph a linear equation in two variables, we first need to obtain some ordered pair solutions. Let's return to our above example. So far, we know that (1,-6) is a solution. We only need two points to make a line, but three is recommended to guard against errors. In your class, you will probably see some sort of table set up with ordered pairs:

Let's complete our table by generating two additional ordered pairs. Again we do this by choosing a value for either x or y and solving for the other.
Let's pick 6 for y:

-2x - 6 = 4

-2x = 10

-2/-2 x = 10/-2

x = -5

(-5,6)

Let's update our table:

Let's find one additional point. Suppose we pick -1 for x:

-2(-1) - y = 4

2 - y = 4

-y = 2

-1/-1 y = 2/-1

y = -2

(-1,-2)

Let's update our table:

Now that we have three ordered pairs: (1,-6), (-5,6), and (-1,-2), we can graph our equation. The first step is to
plot the ordered pairs on the coordinate plane:
Now we can connect the points with a line. Each point on the line represents a solution to the equation.
We place arrows at each end of the line to indicate that the line continues in each direction forever. This also shows that there is an infinite number of solutions for
our equation:
Let's try an example.

Example 1: Graph each line

x + 3y = 6

Let's first get three ordered pairs. Again, we can choose any value we want for x and solve for y, or vice versa. Since graphing is much easier with integer values, we want to choose values that will produce integers. Let's start out with three chosen values for x:

-3, 0, and 3:

(-3,_), (0,_), (3,_)

Let's find our missing y value for each case:

(3,_)

3 + 3y = 6

3y = 3

3/3 y = 3/3

y = 1

(3,1)

(0,_)

0 + 3y = 6

3y = 6

3/3 y = 6/3

y = 2

(0,2)

(-3,_)

-3 + 3y = 6

3y = 9

3/3 y = 9/3

y = 3

(-3,3)

Let's update our table:

We can plot our ordered pairs:

(3,1), (0,2), and (-3,3) and draw a line through the points:

-2x - y = 4

x-intercept: (_,0)

-2x - 0 = 4

-2x = 4

-2/-2 x = 4/-2

x = -2

x-intercept: (-2,0)

If we look at our graph again, we can see that we cross the x-axis at the point (-2,0): Similarly, when the graph crosses the y-axis, the x-coordinate will be zero. This means we can find our y-intercept by plugging in a 0 for x and solving for y. Let's again use the equation:

-2x - y = 4

y-intercept: (0,_)

-2(0) - y = 4

-y = 4

-1 • -1y = 4 • -1

y = -4

y-intercept: (0,-4)

If we look at our graph, we can see that we cross the y-axis at the point (0,-4): Let's take a look at an example.

Example 2: Find the x-intercept and y-intercept

4x - 5y = -20

To find the x-intercept, we plug in a 0 for y and solve for x:

x-intercept: (_,0)

4x - 5(0) = -20

4/4 x = -20/4

x = -5

x-intercept: (-5,0)

To find the y-intercept, we plug in a 0 for x and solve for y:

4(0) - 5y = -20

-5y = -20

-5/-5 y = -20/-5

y = 4

y-intercept: (0,4)

x = -5

When observing the equation, it only appears to have one variable (x). Although this is true, we can still write the equation with two variables as:

x + 0y = -5

This trick of using 0 as the coefficient will be used a lot moving forward. Thinking about the equation, no matter what we choose for y, the value for x is always -5. This creates a vertical line. Let's show three points using a table:

We can use these points as a guide to graph our equation. Really we can just go to x = -5 on the coordinate plane and draw a vertical line:
Lastly, we will also see a horizontal line. As an example, suppose we see the following equation:

y = 4

Again, we don't see an x variable present, but we can re-write our equation as:

0x + y = 4

For any value of x, y will always be 4. Again, we can create a table of values with any three numbers picked for x, y will always be 4. The quicker method is just to go to y = -4 on the coordinate plane and draw a horizontal line. Let's set up a table where x = 6, 0, and -6:

Let's sketch the graph of our equation:
Let's look at an example.

Example 3: Graph each equation

x = 7

To graph this equation, we can go to x = 7 on the coordinate plane and draw a vertical line:

-2x - y = 4

We write a solution for a linear equation in two variables as an ordered pair (x,y). When we encounter a linear equation in two variables, there is an infinite number of solutions. To obtain a solution, we can choose a value for x and solve for the unknown y. Let's suppose we take our above equation and choose a value of 1 for x. This gives us the ordered pair:

(1, __)

The blank is the unknown y value. We can just plug in a 1 for x in our equation and simplify:

-2(1) - y = 4

-2 - y = 4

-y = 4 + 2

-y = 6

-1/-1 y = 6/-1

y = -6

This tells us our unknown y value is (-6). This means our ordered pair is:

(1,-6)

We can check by plugging in a 1 for x and a (-6) for y:

-2(1) - (-6) = 4

-2 + 6 = 4

4 = 4

Since we can make as many ordered pair solutions for a linear equation in two variables as we would like, it is normal to graph the equation. This allows us to get a visual representation of all solutions for the equation. To graph a linear equation in two variables, we first need to obtain some ordered pair solutions. Let's return to our above example. So far, we know that (1,-6) is a solution. We only need two points to make a line, but three is recommended to guard against errors. In your class, you will probably see some sort of table set up with ordered pairs:

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

1 | -6 |

_ | _ |

_ | _ |

-2x - 6 = 4

-2x = 10

-2/-2 x = 10/-2

x = -5

(-5,6)

Let's update our table:

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

1 | -6 |

-5 | 6 |

_ | _ |

-2(-1) - y = 4

2 - y = 4

-y = 2

-1/-1 y = 2/-1

y = -2

(-1,-2)

Let's update our table:

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

1 | -6 |

-5 | 6 |

-1 | -2 |

Example 1: Graph each line

x + 3y = 6

Let's first get three ordered pairs. Again, we can choose any value we want for x and solve for y, or vice versa. Since graphing is much easier with integer values, we want to choose values that will produce integers. Let's start out with three chosen values for x:

-3, 0, and 3:

(-3,_), (0,_), (3,_)

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

3 | _ |

0 | _ |

-3 | _ |

(3,_)

3 + 3y = 6

3y = 3

3/3 y = 3/3

y = 1

(3,1)

(0,_)

0 + 3y = 6

3y = 6

3/3 y = 6/3

y = 2

(0,2)

(-3,_)

-3 + 3y = 6

3y = 9

3/3 y = 9/3

y = 3

(-3,3)

Let's update our table:

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

3 | 1 |

0 | 2 |

-3 | 3 |

(3,1), (0,2), and (-3,3) and draw a line through the points:

### How to Find the x-intercept and y-intercept

When we start graphing equations, we will often find it useful to know where the graph crosses through the x-axis and y-axis. The point where the graph crosses the x-axis is known as the x-intercept. When the graph crosses the x-axis, the y-coordinate will always be zero. This means we can find our x-intercept by plugging in a 0 for y and solving for x. Let's use our earlier equation of:-2x - y = 4

x-intercept: (_,0)

-2x - 0 = 4

-2x = 4

-2/-2 x = 4/-2

x = -2

x-intercept: (-2,0)

If we look at our graph again, we can see that we cross the x-axis at the point (-2,0): Similarly, when the graph crosses the y-axis, the x-coordinate will be zero. This means we can find our y-intercept by plugging in a 0 for x and solving for y. Let's again use the equation:

-2x - y = 4

y-intercept: (0,_)

-2(0) - y = 4

-y = 4

-1 • -1y = 4 • -1

y = -4

y-intercept: (0,-4)

If we look at our graph, we can see that we cross the y-axis at the point (0,-4): Let's take a look at an example.

Example 2: Find the x-intercept and y-intercept

4x - 5y = -20

To find the x-intercept, we plug in a 0 for y and solve for x:

x-intercept: (_,0)

4x - 5(0) = -20

4/4 x = -20/4

x = -5

x-intercept: (-5,0)

To find the y-intercept, we plug in a 0 for x and solve for y:

4(0) - 5y = -20

-5y = -20

-5/-5 y = -20/-5

y = 4

y-intercept: (0,4)

### Graphing Vertical Lines and Horizontal Lines

We have two special case scenarios that occur when graphing linear equations in two variables. Let's suppose we encounter an equation such as:x = -5

When observing the equation, it only appears to have one variable (x). Although this is true, we can still write the equation with two variables as:

x + 0y = -5

This trick of using 0 as the coefficient will be used a lot moving forward. Thinking about the equation, no matter what we choose for y, the value for x is always -5. This creates a vertical line. Let's show three points using a table:

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

-5 | -6 |

-5 | 0 |

-5 | 6 |

y = 4

Again, we don't see an x variable present, but we can re-write our equation as:

0x + y = 4

For any value of x, y will always be 4. Again, we can create a table of values with any three numbers picked for x, y will always be 4. The quicker method is just to go to y = -4 on the coordinate plane and draw a horizontal line. Let's set up a table where x = 6, 0, and -6:

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

-6 | 4 |

0 | 4 |

6 | 4 |

Example 3: Graph each equation

x = 7

To graph this equation, we can go to x = 7 on the coordinate plane and draw a vertical line:

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