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Ordered Pair

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Two elements a b and listed in a specific order form an ordered pair, denoted by (a, b). In an ordered pair (a, b); a is regarded as the first element and bis the second element. 

It is evident from the definition that

  1. (a, b) $\neq$ (b, a)
  2. (a, b) = (c, d), iff a = c, b = d
Equality of Ordered Pair

Two ordered pairs ($a_1$, $b_1$) and ($a_2$, $b_2$) are equal iff

$a_1 = a_2$ and $b_1 = b_2$

i.e., ($a_1$, $b_1$) = ($a_2$, $b_2$)

⇒ $a_1 = a_2$ and $b_1 = b_2$

Thus, it is evident from the definition that (1, 2) $\neq$ (2, 1) and (1, 1) $\neq$ (2, 2)

Sample Problem 1

If $\left(\frac{x}{3} + 1, y - \frac{2}{3}\right)$, find the values of x and y.
(a) 2, 1
(b) -2, 1
(c) 2, 2
(d) 1, 1

Answer: (a)

Given, $\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right)$

⇒ $\frac{x}{3}$ + 1 = $\frac{5}{3}$ and y - $\frac{2}{3}$ = $\frac{1}{3}$

⇒ $\frac{x}{3}$ = $\frac{5}{3}$ - $\frac{1}{1}$ and y = $\frac{2}{3}$ + $\frac{1}{3}$

⇒ $\frac{x}{3}$ = $\frac{5-3}{3}$ and y = $\frac{2+1}{3}$

⇒ $\frac{x}{3}$ = $\frac{2}{3}$ and y = $\frac{3}{3}$

⇒ So, x = 2, y = 1

Sample Problem 2

Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d, i.e., (a, b) = (c, d). Find the values of x and y, if (2x - 3, y + 1) = (x + 5, 7)

Answer:

We will solve by equality of ordered pairs

Given 2x - 3 = x + 5 and y + 1 = 7

⇒ 2x - x = 5 + 3 

⇒ x = 8

and y = 7 - 1 

⇒ y = 6

Hence x = 8 and y = 6.

Note: Both the elements of an ordered pair can be the same i.e., (2, 2), (5, 5).


Sample Problem 3

1. Ordered pairs (x, y) and (2, 7) are equal if x = 2 and y = 7.

2. Given (x - 3, y + 2) = (4, 5), find x and y.

Answer:

(x - 3, y + 2) = (4, 5)

⇒ x - 3 = 4 and y + 2 = 5

Then x = 4 + 3 and y = 5 - 2 or x = 7 and y = 3

Sample Problem 4

Given (6a, 6) = (4a - 8, b + 2)

Answer:

(6a, 6) = (4a - 8, b + 2)

Then, 6a = 4a - 8 and 6 = b + 2

⇒ 6a - 4a = 8 and b = 6 - 2

⇒ 2a = 8 and b = 4

⇒ a = 4

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