Ordered Pairs

Ordered Pair definition

Simply put, an ordered pair could be defined as a combination of two numbers from the x-axis (abscissa) and the y-axis (ordinate) written in a specific order enclosed within parentheses and separated by a comma.

It is used to represent two variables, one on the x-axis and another on the y-axis. Furthermore, it helps to locate a point on the Cartesian plane for better visual understanding.

As the name suggests, in ordered pairs, the order of the points is very important. It represents a specific and unique point in a coordinate plane.

An ordered pair of points such that x=3 and y=4 denoted by (3,4) is different from points x=4 and y=3 denoted by (4, 3) as the position of these changes when plotted as a coordinate point.

Hence, unless x is not equal to y, (x, y) and (y, x) are two different ordered pairs with a different position in a two-dimensional plane.

Ordered pair meaning

An ordered pair is denoted by writing the specific point of an axis in a particular order enclosed within parentheses. Ordered pairs are also known as 2-tuples, or sequences of length 2.

A simple representation of an ordered pair is (x, y) where x denotes a point in the x-axis and y denotes a point in the y-axis.

Ordered Pair = (x, y)

Here, x = abscissa, which is a measure of projection of point on the primary axis, whose absolute value is the distance between the projection and the origin of the axis, and whose sign is given by the location on the projection relative to the origin.

 y = ordinate, which is the measure of projection of point on the secondary axis, whose absolute value is the distance between the projection and the origin of the axis, and whose sign is given by the location on the projection relative to the origin.

The number of x which corresponds to the x-axis is called the x coordinate and the number which corresponds to the y-axis is called the y coordinate.

For example, in the ordered pair (3, 6), 3 is the x coordinate and 6 is the y coordinate. 3 can also be called the first element and 6 the second element.

The numeric value of ordered pairs could be integers and fractions.

Ordered pair examples

Let’s see an example of an ordered pair and how to plot them on a cartesian plane. The points (2, 2), (2, -2), (-2, 2) and (-2, -2) are all different points of a coordinate plane and when plotted and joined forms a square. Thus, ordered pairs form the base of different geometrical shapes that could be plotted on a cartesian plane and hence help in a better understanding through visual aid.

Another example of ordered pairs is (5, 6) and (6, 5). As explained earlier, these two points are different from each other and this difference is because of their order. To understand it better, imagine these two points on a cartesian plane.

The position of the point (5, 6) is not the same as the position of the point (6, 5).

One of the most important things to remember in ordered pairs is the order or sequence of the numbers enclosed within the parenthesis.

Properties of ordered pairs

1)Equality of ordered pairs

Two ordered pairs are said to be equal if the coinciding first elements or x coordinates and the coinciding second elements or y coordinates of the two ordered pairs are equal.

This means, if (a, b) and (m, n) are two ordered pairs such that a=m and b=n, the (a, b) = (m, n)

For example, A(8, 9) and P(8, 9) are two ordered pairs such that A is equal to P.

Here is a typical example of a sum based on the property of equality of ordered pairs.

1. Given are two ordered pairs such that (a, b) = (c, d) such that a=c and b=d. Find the values of x and y such that (2x, 7) = (16, y-3)

Solution- According to the equality of ordered pairs,

    we have, 2x=16

                x= 16/2= 8

         and, 7= y – 3

                y= 7+3= 10

Therefore, x=8 and y=10

Questions based on ordered pairs

1. Are the ordered pairs (3, 6) and (6, 3) equal?

Solution- No, according to the property of equality of ordered pairs, two ordered pairs are equal if and only if the corresponding first components are equal and corresponding second components are equal.

Since 3 is not equal to 6 and 6 is not equal to 3, the ordered pairs are not equal.

2. Determine the abscissa and ordinate of the ordered pairs (3, 6) and (7, 2)

Solution- Abscissa is the x coordinate which is also known as the first element in ordered pairs.

Hence, Abscissa in the ordered pairs is 3 and 7.

The Ordinate is the y coordinate which is also known as the second element in ordered pairs.

The Ordinate in the ordered pairs is 6 and 2.

Conclusion

Ordered pairs are an essential concept of coordinate geometry. Questions based on coordinate geometry are frequently asked in competitive exams  and having a good grip on the basics could help aspirants a great deal.

Ordered pairs are used for better visual comprehension on a coordinate plane and help form geometric shapes and solve word problems. The order of the ordered pairs cannot be changed unless the value of the x and y coordinates are the same.