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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to see how useful the Cartesian coordinate system is. Learn how points are labeled and how you can plot points and graph lines using the system.

The **Cartesian coordinate system** is a system that tells you your exact position on a graph. Why is this useful? It is useful because you can use this system to graph a series of points that will provide solutions to your equations. The equations that are used with this system will have two variables, one for each axis.

The Cartesian coordinate system uses a horizontal axis that is called the *x*-axis and a vertical axis called the *y*-axis. Equations for lines in this system will have both the *x* and *y* variable. For example, the equation 2*x* + *y* = 2 is an example of a line in this system. Lots of equations can be written with these two variables. You can have equations that explain population growth and you can have equations that explain your cell phone bill.

What does a Cartesian coordinate system look like? It looks like graph paper with a dark line for the *x*-axis and another dark line for the *y*-axis. These two axes cross perpendicular to each other. The point where they cross is at point 0 for each axis. This graph extends forever in all directions. Both axes are labeled with numbers according to the number line.

You can think of the *x*-axis as a horizontal number line and the *y*-axis as a vertical number line. When you draw one, though, you only draw it as big as you need it to be. If you only need the *x*-axis and the *y*-axis to go up to 10, then you can stop there for both axes. Let's look at some examples to see how it works. Let's start with points.

Before plotting points, let's talk about how points are labeled. Points are labeled based on their position on the graph in relation to both the *x*-axis and the *y*-axis. For example, if you look at this point and you draw an imaginary vertical line straight down, you see that it lands where the *x*-axis equals 2. This *x*-axis number will be the first part of your label.

Drawing an imaginary horizontal line, you see that it crosses the *y*-axis where it equals 3. This is the second part of the label for that point. To label this point, then, you write the first part and second part separated by a comma inside a pair of parentheses like this: (2, 3).

Look at this second point. Try labeling this one on your own. Where does the imaginary vertical line cross the *x*-axis? What about the imaginary horizontal line?

Because the imaginary vertical line crosses the *x*-axis at 4 and the imaginary horizontal line crosses the *y*-axis at -2, the point is labeled (4, -2).

Now that you know how points are labeled, plotting points becomes easier. Plotting points is essentially labeling points in reverse.

For example, if you are given the point (0, 3) to plot, you would first find the vertical imaginary line where the *x* value is 0 and then you would follow this line either up or down until you reach the horizontal imaginary line where the *y* value is 3. Once you've found it, place a dot on the graph and label it with (0, 3).

You try plotting the point (-1, 2). You first find the vertical imaginary line where the *x*-axis is -1. Once you've found it, you follow it up to where the *y*-axis is 2. You have found the point. Now place a point there and label it with (-1, 2). You did it!

To graph lines, we first calculate two or more points and then we plot them. There are two ways you can graph a line. You can create a table of two or more points or you can calculate the *x*- and *y*-intercepts to plot. The intercepts are where the imaginary lines from the point cross each axis.

Let's try graphing the line 2*x* + *y* = 2 using both the table method and the intercept method. For the table method, we first create a table with two columns, one for the *x*-axis and one for the *y*-axis like this:

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

We can choose to calculate just two points or more. I recommend calculating three points so you can be sure you have a straight line. To calculate them, you set the *x* value and then find the *y* value using algebra.

The easiest numbers to choose for the *x* value are -1, 0, and 1. I recommend using these. Let's see what happens when we plug in these numbers into our equation of a line. This is what happens when *x* = -1:

2(-1) + *y* = 2

-2 + *y* = 2

-2 + 2 + *y* = 2 + 2

*y* = 4

So our first point is (-1, 4). We can fill our table like this:

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

-1 | 4 |

Moving on, this is what happens when *x* = 0:

2(0) + *y* = 2

0 + *y* =2

*y* = 2

Our table now looks like this:

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

-1 | 4 |

0 | 2 |

For *x* = 1, this is what happens:

2(1) + *y* = 2

2 + *y* = 2

2 - 2 + *y* = 2 - 2

*y* = 0

Our finished table now looks like this:

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

-1 | 4 |

0 | 2 |

1 | 0 |

Now we can go and plot these three points on our graph. Plotting these, we see that they do follow a straight path. We then draw a straight line through these points for our line.

The second method, the intercept method, uses only two points. To calculate the points using this method, we first set the *x* to 0 to calculate our *y* for our first point, and then we set the *y* equal to 0 to calculate the *x* for our second point. Let's see how it works.

When *x* = 0, we get this:

2(0) + *y* = 2

0 + *y* = 2

*y* = 2

So, our first point is (0, 2). Next, when *y* = 0, we get this for *x*:

2*x* + 0 = 2

2*x* = 2

2*x* / 2 = 2 / 2

*x* = 1

So our second point is (1, 0). Plotting these two points and drawing a line through the two points, we get a line like this.

This line should look just like the line we just drew since we are graphing the same line.

Unless the problem tells you otherwise, you can use either method to graph your line. Choose the one that is easier for you. But I recommend practicing both methods so you get a good feel for both. Use the equation form that we just used, but try different numbers to see what kinds of lines you get.

What have we learned? We've learned that the **Cartesian coordinate system** is a system that tells you your exact position on a graph. It has a horizontal axis called the *x*-axis and a vertical axis called the *y*-axis. Points are labeled according to where they are located in relation to these two axes. By drawing imaginary horizontal and vertical lines from the point, we find where the point crosses each axis. These numbers form the point's label.

For example, the point (2, 1) has imaginary lines that cross the *x*-axis at 2 and the *y*-axis at 1. The *x*-axis number is always written first, followed by the *y*-axis number. Equations for lines will include both the *x* and *y*. 2*x* + *y* = 2 is an example of an equation of a line. You can see that it has both variables in the equation. To graph this line, you can use the table method or the intercept method.

The table method involves choosing at least two values for the *x* and finding out what the *y* equals for those values and then plotting the points. Three values are recommended to ensure a correct and straight line. The intercept method involves setting *x* equal to 0 to find *y* for the first point and then setting *y* equal to 0 to find *x* for the second point. The two points are then graphed and a line drawn through them.

At the end of this lesson, you should be able to:

- Explain the Cartesian coordinate system
- Locate points on a graph
- Solve line equations using either the table method or the intercept method and graph a line

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Geometry: High School15 chapters | 160 lessons

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