Cartesian coordinate system is a two/three dimensional plane with two/three axes. In two dimensional plane there are two axes, a horizontal axis, known as the x-axis and a vertical axis, known as the y-axis used to plot points and line graphs. It was invented by the French Mathematician Rene Descartes. Each point is represented by an ordered pair (x, y). The two axis splits the coordinate system into four quadrants.
Any Point in the first quadrant will be positive for both x and y.
In second quadrant the points will have negative x and positive y.
Any Point in the third quadrant will be negative for both x and y.
The fourth quadrant contains points with positive x and negative y coordinates.

In a three dimensional space choose an ordered triplet of lines where any two of them can be perpendicular, a single unit of length for all the three axes. Three dimensional coordinate system can have multiple origins and can define cubes, spheres,etc.,
Polar Coordinates
In the image we see that a point Q is referred by three real numbers indicating the positions of the perpendicular projections by three fixed lines called the axes which intersect at the origin.

Solved Examples

Question 1: Graph the equation 3 + 5y = x
5y = x -3
           y = $\frac{x}{5}$ - $\frac{3}{5}$
Cartesian Coordinate System Example

From the graph we see that for x = 8 the y intercept is y = 1. The point is (8, 1)
In the given equation 3 + 5y = x , when y = 1 we get x = 8. The point is (8, 1).

Question 2: For the points A = (2, 9) and B = (7, 2) find the distance between the Cartesian coordinates.
The formula to find the distance between the cartesian coordinates is given by

D(x, y) = $\sqrt{(x_{2}-x_{1})^{2}+ (y_{2}-y_{1})^{2}}$
= $\sqrt{(7-2)^{2}+ (2-9)^{2}}$
= $\sqrt{(5)^{2}+ (-7)^{2}}$
= $\sqrt{(25) +( 49)}$

= $\sqrt{74}$

Question 3: Find the midpoint of the line segment joining X( -9, 3) and Y(7, 6).
Let ($x_{1}$, $y_{1}$) = ( -9, 3) and ($x_{2}$, $y_{2}$) = (7, 6)
The midpoint formula is = $\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}$

Plugging in the given points we get ($\frac{-9+7}{2}, \frac{3+6}{2}$)
= (-1, 4.5)
Therefore (-1, 4.5) is the line segment joining X( -9, 3) and Y(7, 6).