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Ordered pairs are a pair number written in a certain order. It is used to locate the position on a graph in the coordinate system, where the "x" (horizontal) value is first, and the "y" (vertical) value is second. In this article, we will explore the examples, properties, and applications of ordered pairs.
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What are Ordered Pairs?
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A pair of numbers that are written in a particular order, enclosed in brackets are known as ordered pairs. It is a combination of X and Y coordinates that has two values written in a particular order within brackets.
Ordered Pair = (x, y)
where, x = abscissa, the distance of a point from the x-axis.
and, y = ordinate, the distance of a point from the y-axis.
While plotting this, we need to draw a dot at the coordinates that correspond to the ordered pair. The x coordinate is the distance we need to count from the x-axis, and that of the y coordinate is the distance from the y -axis.
Check out notes for Relations and Functions
Ordered Pair Example
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Ordered Pair = (2, 8)
This is the ordered pair with two integers, with 2 as the abscissa, and 8 as the ordinate.
The ordered pair (2,8) is not the same as the ordered pair of (8,2), where the abscissa and ordinate are 8,2 respectively.
Set of Ordered Pairs
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Ordered pairs are the pair of numbers written in a fixed order, within brackets. If 'a' and 'b' are two elements, then the ordered pairs (a, b) and (b, a) are not the same. In the ordered pair (a, b), the first element is a, and the second element is b.
If A and B are two sets, such that a ∈ A and b ∈ B, then by the ordered pair of an element, we mean that ‘a’ is the Ist element and ‘b’ is the IInd element of the ordered pair.
If in the ordered pair, the position of the elements is changed, then the ordered pair also gets changed. Hence, (a, b) ≠ (b, a).
Equality of Ordered Pairs
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The property of equality of ordered pairs is:
- Equality of Ordered Pairs
Any two ordered pairs are said to be equal if the first and second corresponding elements are equal.
For example:
Consider two ordered pairs (a, b) and (c, d). These two ordered pairs will be equal only if a=c and b=d, i.e., (a,b) = (c,d).
Note: Both the elements of an ordered pair can be the same i.e., (2, 2), (5, 5).
Applications of Ordered Pairs
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The ordered pairs has many applications:
- It is used in data comprehension and word problems and statistics.
- It is used in coordinate geometry to represent geometric objects and figures in an open space to aid visual comprehension.
- Ordered pairs are used to represent the center, vertices, and length of the sides with coordinates of the square, circle, rectangles, and polygons.
Things to Remember based on Ordered Pairs
- A pair of numbers that are written in a particular order, enclosed in brackets are known as ordered pairs.
- It can be represented as, Ordered Pair = (x, y)
- If the position of the elements of the ordered pair is changed, then the ordered pair also changes
- It is applied in statistics, to depict the center, vertices of a circle, square, rectangle, etc.
- Ordered pairs are used for data comprehension and to aid visual comprehension.
Sample Question based on Ordered Pairs
Ques. Write down a property of the Ordered pair? (1 mark)
Ans. The property of equality in ordered pairs is that if (a, b) and (c, d) are equal when a = b, and c = d.
Ques. If (x, y) = (-1, 4), find the value of 3x + 2y - 4? (2 marks)
Ans. Given x = (-1) and y = 4.
Substituting the value of x, y in the given equation:
3x + 2y - 4 = 3(-1) + 2(4) - 4 = 1.
Ques. If A = {2,4,6,9} and B = {4,6,18,27,54}, and we are given a and b such that a ∈ A, b ∈ B, find out the set of ordered pairs such that a is factor of b and a<b. (2 marks)
Ans. The sets are A = {2, 4, 6, 9} B = {4, 6, 18, 27, 54},
Now we need to pair the sets such that the a is a factor of b and a < b .
Therefore, we find that the ordered pairs are:
{(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)}
Ques. Plot the point “P” with coordinates 6, 4. (3 marks)
Ans. As per our understanding of ordered pairs, we can write the point P as:
P = (6, 4)
The first number of the ordered pair is 6, which shows the distance from the X-axis. And the second number 4, shows the distance from the Y-Axis.
Move 6 steps toward the right from the origin along the x-axis, and from there, take 4 steps up. Then mark that point as (6,4)
Ques. Is there any difference between the ordered pairs (6, 4), (4, 6)? (2 marks)
Ans. Yes, there is a difference between the ordered pairs (6, 4), (4, 6) as demonstrated below. If the position of the ordered pair elements is changed, the position of the points also changes.
Ques. Find the values of x and y, if (2x - 3, y + 1) = (x + 5, 7). (3 marks)
Given: 2x - 3 = x + 5 and y + 1 = 7
⇒ 2x-x = 5+3
⇒ x = 8
and y = 7 - 1
⇒ y = 6
Hence, we get
x = 8
y = 6
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