Cartesian Coordinate System

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Cartesian coordinates

The Cartesian coordinate system relies on points on a grid. These points are the same theoretical points we know and love in geometry, only they are fixed to exact spots on a numbered network of horizontal and vertical lines. The points in geometry and Cartesian coordinates have no dimension; they mark only a position in space.

The Cartesian coordinate system, named after French mathematician and philosopher René Descartes, relies on points on a grid. Descartes thought up the system in 1637, supposedly after seeing a fly on his ceiling.

Many of you are lucky enough to live in cities with street grids, in places like Atlanta, Houston, Los Angeles, Miami, and Portland. Those neatly ordered grids allow drivers to move along a left-right street, turn onto a north-south street, and find just about any location. Street grids are excellent, real-life applications of Cartesian coordinate systems.

The Cartesian coordinate system

Consider the horizon of a vast desert. Imagine that the horizon is thought of as 0, and you can trace parallel lines above that horizon line, each exactly the same distance apart as all the rest. Those horizontal lines, stacked up and down (even below the horizon line), would be steps up and down the y-axis of a two-axis Cartesian coordinate system.

X and y axis

The y-axis lines run left and right but measure distance up and down, away from the 0 line (our horizon). So a y-axis spot 3 units above 0 would be somewhere on a horizontal line three distance units up (whatever our distance units are, like 1s, 10s, 100s and so on). You can also have points on lines below 0, using negative y-axis numbers. The y-axis measures up-down changes greater and less than 0.

Vertical lines, lines at right angles to those horizontal lines, would be the x-axis of your system. Though x-axis lines run up and down, they measure distances to the left and right of the origin, or center of the coordinate system. A point at +4 on the x-axis will lie somewhere on a vertical line four distance units to the right of the origin. The x-axis measures left-right changes greater and less than 0.

Cartesian graph

Look at this grid. See if you can find which labeled point is at (3, 5):

[insert grid of Quadrant 1 only with labeled points: F (2, 6); R (3, 5); O (4, 4); and G (5, 3)]

Point R is at (3, 5), meaning we moved three spaces to the right of x-axis 0, then five spaces up to the line at y-axis 5.

Ordered pairs

To navigate along the x-axis and y-axis, everyone needs to agree on the order of the identifying intersections. All ordered pairs, or sets of identifying numbered points, are given first as the x-axis value and then as the y-axis value, like this: (x, y).

Here is an ordered pair and its meaning:

In our drawing above, Point R is not at (5, 3), because the x-axis value is 5, not 3. We did not go far enough to the right to be at an x value of 5. Point G is at (5, 3), or 5 moves to the right along the x-axis, then 3 moves up to the y value.

Did you say (2, 6), (3, 5), (4, 4), and (5, 3)? We sure hope so, because those are the correct ordered pairs.

The easiest way to navigate a Cartesian coordinate system is to first move along the x-axis to find the first position, then up to the y-axis position. So, for our four points F, R, O and G, you can hop to it by moving first to the x value, then up or down to the y value. If you are asked not to find an existing point but to plot one, you do the same thing.

Plot Point S on the same Cartesian grid. Point S must be at (6, 2). Think about what that means: move six units to the right along the x-axis, then two units up to the y value. It should be a point that continues the line already created, so now you have a line of FROGS.

Graphing lines

You plot points on a Cartesian coordinate system in order to graph lines. By plotting as few as two points, you can graph a line they create ("Two points determine a line").

You have two ways to graph lines on a grid system:

1. The table method

2. The x- and y-intercepts method

The table method refers to a mathematical table, not one you sit around. With the table method, you have two columns representing given x values and y values. The intercepts method refers to the imaginary lines created on your grid cross the x-axis and y-axis.

We will try both methods with a very simple equation, $x+y=8$.

The table method

The two columns of the table method reflect the x values and y values:

Pick any number for your x value and put it into your x value column. Then, think, algebraically, what is needed to solve for y? Start with something easy, like 0 and then 1:

You can use algebra, though the steps should be easy to do mentally:

And...

We can see the y values need to be 8 and 7, and if we continue to plug in numbers for x, we get predictable results:

On a Cartesian coordinate system, those points line up like this:

[insert Quadrant I grid with those points, which are the same as the FROG points, and connect them with a line; this might do better as an animation that plots the points and then draws the line]

X and y intercepts method

Plotting a line with the x-intercept and y-intercept is just about as easy as the table method. Set x to 0 for the first point; set y to 0 for the second point. Just as a reminder, our equation was $x+y=8$.

With $x=0$ you would have:

And with $y=0$ you would have:

Our first point will be (0, 8) and our second point will be (8, 0), and with those two points, we can create a line.

If that line and the line from the table method look like the line from our FROGS, you are correct. It is the same line, created using the equation $x+y=8$.

Applications

The Cartesian coordinate system finds uses in maps, statistics, graphing, and calculus. If you look around, you will find the concept applied in many branches of mathematics and science. Even games and sports use the idea, like orienteering, chess, and the game of Battleship.

Examples

Use your own graph paper to try these challenges. Remember to use either method (or both, to check yourself) and to work through each ordered pair carefully.

1. $y=-x$

2. $2x+y=-10$

3. $x+2y=4$

For these, use graph paper to plot the points and then answer each question:

1. Are the points (-6,6), (6,6), (6,-6) and (-6,-6) on a straight line?

2. What shape do the points (-6,6), (6,6), (6, -6), and (-6,-6) create?

3. Describe the line created by these points: (-8,2) and (8,2).

4. Does the point (0,2) lie on that line?

5. Does the point (-4,2) lie on that same line? How do you know?

6. Identify a point on a line created by the points (0,10) and (0,-7).

Here are the answers, but try not to peek until you do the work!

1. The points are not on a straight line, because the ordered pairs do not all have identical x values or y values.

2. The shape the four points create is a square.

3. The line created by the two points is a horizontal line.

4. Point (0,2) is on the line, because its y value is the same as the other y values.

5. Point (-4,2) is also on the line, for the same reason.

6. A point on a line created by (0,10) and (0,-7) can be any y value but will be along the y axis line, since the x value is 0.

Lesson outcomes

Now that you have gone through this lesson, you are able to explain and sketch out the Cartesian coordinate system. You are also able to locate points on a Cartesian coordinate system and its related organizers like line graphs, maps, and graphs of functions in calculus, algebra and geometry. Additionally, you are able to graph a line and solve equations of lines using a table, or using the intercept method.