There is an important branch of mathematics, known as **coordinate geometry**. It studies about the position of objects in space and problems based on that. An object or a point is defined by means of coordinates in the space. There are various coordinate systems that represent the position of an object. These coordinate systems are - Cartesian coordinate system, spherical coordinate system, cylindrical coordinate system and polar coordinate system. Cartesian coordinate system is most commonly used in mathematics. This system is in two or three dimensions and even higher.

In the two-dimensional Cartesian coordinate system, we define position of a point with the help of ordered pair. We can define the ordered pairs as coordinates in the graph. In math, the ordered pairs are combined with two quantities and it is represented in a certain order. An ordered pair is defined as (x , y). The parenthesis symbol is used for ordered pair representation. The first element in ordered pair i.e. x element, is known as **abscissa**; while second element or y element is termed as **ordinate**. The abscissa is the position of a point along the X axis and ordinate is its position in the direction of Y axis. The ordered pair defines the coordinates or position of a point in two dimensions. Now, we are going to see about ordered pairs.

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The ordered pair is a representation of two objects in parentheses with some order. The ordered pair illustrates the coordinates of point in graph and the notation of ordered pair is (a, b).

In ordered pair notation, the left side value â€˜aâ€™ is x coordinate that is x-axis value and the right side value â€˜bâ€™ is y coordinate that is y-axis value. The ordered pairs may be positive or negative.The ordered pairs are (a_{1}, b_{1}) and (a_{2}, b_{2}).

(a_{1}, b_{1}) = (a_{2}, b_{2}) if and only if a_{1} = a_{2} and b_{1} = b_{2}.

The set of all second elements of any order ordered pair is called the range of the ordered pair. So, if we have {(1,a), (2,b), (3, c)} as the set of ordered pairs, then the set {a, b, c} is the range of the ordered pair. If we have a graph of any ordered pairs, then the range of the ordered pair is those y values which are present in the graph.

If we have any equation in two variables or more than two variables, then we can make a tabular data for that equation and get the solution of that equation with the help of the table in terms of ordered pairs.

For example, if we have y = x + 3, then we can make a table for it and get the solution of this equation. In the table, first we make four column for x, x + 3, y and for ordered pairs respectively. After that, put the values of x and get the corresponding values of y. Then, arrange these values in terms of ordered pairs.

The ordered pairs are plotted in graph based on quadrants.

- The first quadrant is representing the ordered pairs like (+, +).
- The second quadrant is representing the ordered pairs like (-, +).
- The third quadrant is representing the ordered pairs like (-, -).
- The fourth quadrant is representing the ordered pairs like (+, -).

As per the given ordered pair, we can plot the points in the graph and define it as (a,b). The first value a lies on x-axis and the second value b lies on y-axis.

Here, the x coordinate is 7 and the y coordinate is 6.

Here, the x coordinate is -9 and the y coordinate is -5.

We know that a set of ordered pairs is a relation and a function is a relation in which any two ordered pairs does not contain the same first element. i.e. we can say that for every element of y, there is a unique element of x. For any function, the first element of the ordered pair is known as the input and the second element is known as output.

- The set of ordered pairs (a, 1), (b, 2), (c, 3), (d, 4) is a function because there is a unique element of x for every element of y.
- The set of ordered pairs (1,7), (2, 8), (1, 4), (3, 6) is not a function since there are two pairs with the first element 1.

To find the solution of an equation y = 3x + 1, we can put the values of x as 0, 1, 2, 3,.......

Now,

If x = 0, then y = 1

If x = 1, then y = 3 + 1 = 4

If x = 2, then y = 7

If x = 3, then y = 10 etc.

So, we have the solution of the equation in terms of ordered pairs as (0,1), (1, 4), (2, 7), (3, 10), ...... etc. In this type of cases, we have an infinite number of ordered pairs as the solution of the equation.

Given below are some of the examples on ordered pairs.

Given y = x^{2}. Now, put the different values of x to get the set of ordered pairs as the solution of the equation.

If x = 0, then y = 0

If x= 1, then y = 1

If x = 2, then y = 4

If x = 3, then y = 9

If x = -1, then y = 1 etc.

If x = 0, then y = 0

If x= 1, then y = 1

If x = 2, then y = 4

If x = 3, then y = 9

If x = -1, then y = 1 etc.

Then, the solution of the given equation is (0,0), (1,1), (2,4), (3, 9), (-1, 1) etc.

The ordered pair (5, 4) is plotted in first quadrant in graph. Here, the number 5 is x coordinate and the number 4 is y coordinate.

The ordered pair (-5, 5) is plotted in second quadrant in graph. Here, the number -5 is x coordinate and the number 5 is y coordinate.

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