1. The ordered pair is a pair of objects in which the occurrence of each component is governed by some specific rules (order).
  2. The first component of an ordered pair is known as antecedent and the second component is known as consequent.
  3. Two ordered pairs are said to be equal if their corresponding components are equal.

Algebra | Ordered Pairs | Class-09

Introduction

An ordered pair is a pair of objects in which the occurrence of each component is governed by some specific rules (order).

For example: (2,4) is an ordered pair.

The occurrence of 2 and 4 in the ordered pair is governed by a specific rule out of the following rules:

1) y=x²
2) y=2x
3) y=x+2

To find out the rule or the specific order, we need to know at least two ordered pairs.

However, finding the rule of ordered pairs is not our matter.



Rules of writing an ordered pair:

There are a couple of rules to be followed while writing either a single or a set of ordered pairs. One must strictly follow these rules or orders:

1) Every ordered pair is enclosed in parenthesis (small brackets).
2) There shall be only two components (antecedent and consequent) inside an ordered pair.
3) A comma (,) shall separate the two components of the same ordered pair.
4) Every ordered pair shall be written according to the specified rule.

Let us write an ordered pair using the above-mentioned rules with a specified rule 'y = 2x':

1) ( ) [Here, we fulfilled the first rule of enclosing the ordered pair inside the parenthesis.}
2) (x y) [Now, we have written only two components; x 'antecedent' and y 'consequent'. We can neither write more or less than two components.]
3) (x,y) [We have now separated the two components with a comma. It is mandatory.]
4) (1,2) [Now, we follow the rule "y = 2x" where x = 1. Always, the first component is regarded as 'x' and the second component is regarded as 'y']



Components of Ordered Pairs

In an ordered pair, the first component is known as antecedent and the second component is known as consequent.



Equality Of Ordered Pairs:

Two ordered pairs are said to be equal if their corresponding ('x' and 'y') components are equal.

We had already discussed that the first component can also be referred to as 'x' and the last component can also be referred to as 'y'.

When we have two ordered pairs: (a,b) and (x,y). These ordered pairs are said to be equal only if a=x (first components are equal) and b=y (last components are equal).

Some examples of equal ordered pairs: (6-2,4) = (4,3+1), (1,3) = (1,3), (4,5+3) = (2*2,8), and more.

Some examples of inequal ordered pairs: (4,2) and (2,4), (3,4) and (2,6-3), (8,2-1) and (4,5+1).

Key point: If any two order pairs are equal then, their corresponding components are also equal.




Solved Examples:

Which of the following ordered pairs are equal? Write with reason.

a) (2,3) and (4,2) b) (6,4) and (6,5-1)

Answer: a) (2,3) and (4,2). Given ordered pairs are not equal because their corresponding x-components and y-components are not equal.

b) (6,4) and (6,5-1) or, (6,4) and (6,4). Given ordered pairs are equal because their corresponding x-components and y-components are equal.

In each of the following conditions, find the values of x and y.

a) (3x, 2y+2) = (12, 3y-1) b) (x+y, y+3) = (6,2y)

Answer: a) (3x, 2y+2) = (12, 3y-1). From the relation of equal ordered pairs, we know that x and y components of each ordered pairs are equal. So, 3x = 12; i.e. x = 4. And, 2y+2 = 3y-1; i.e. y = 3. Therefore, (x,y) = (4,3).

b) (x+y, y+3) = (6,2y). From the relation of equal ordered pairs, we know that x and y components of each ordered pairs are equal. So, y+3 = 2y; i.e. y = 3. And, x+y = 6; i.e. x = 3. Therefore, (x,y) = (3,3).




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