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This question aims to find the definition of an ordered pair. An ordered pair consists of two coordinates written in a specific order inside the parenthesis, where the x-coordinate is called abscissa and the y-coordinate is called ordinate.
Expert Answer
These ordered pairs are generally used in graphs where they represent the position of points on the graph.
- These ordered pairs make the construction of graphs easy.
- Ordered pairs are used to locate the points on the graph.
Ordered pairs are represented as ($x$,$y$), where the abscissa of the ordered pair is the distance of a point on the x-axis from the origin, and the ordinate of the ordered pair is the distance of a point on the y-axis from the origin.
For example:
An ordered pair $A$= ($4$,$6$) is represented on the graph in the following way, where the value of $x$ is $4$, and the value of $y$ is $6$.
Figure 1
Ordered Pairs in the Cartesian Plane
In a cartesian plane, the point at which the x-coordinate and y-coordinate are zero is called the origin. The distance of a point from the origin determines its numerical value. The x-axis is a horizontal line that determines the value of an independent variable, and the y-axis is the vertical line in a cartesian plane that determines the value of a dependent variable.
Ordered Pairs in a Set
Insets, the abscissa of an ordered pair, is called the first element, and the ordinate of the ordered pair is called the second element. They are represented as:
\[(a,b)\neq (b,a)\]
This expression tells us the importance of order. Changing the order will make $b$ as abscissa and $a$ as ordinate.
Equality of Ordered Pairs
Two ordered pairs ($a$,$b$) and ($c$,$d$) are said to be equal when the corresponding first and second elements of these pairs are equal.
For example:
$a$=$c$ and $b$=$d$ then we will say that, ($a$,$b$)=($c$,$d$).
Numerical Solution
Find the value of the $x$ and $y$ if the given ordered pairs are:
Given: \[(x – 3 , y + 2) = (4 , 5)\]
Required: Values of $x$ and $y$
Equating both ordered pairs give us:
\[x = 4 + 3\]
\[y = 5 – 2\]
\[x = 7\]
\[y = 3\]
Example
Given:
\[(5a – 4 , b + 1) = (3a , 3)\]
Required: Values of $x$ and $y$
\[5a – 4 = 3a\] $and$ \[b + 1 = 3\]
\[5a – 3a = 4\]
\[b = 3 – 1\]
\[b = 2\]
\[2a = 4\]
\[a = 2\]
Image/Mathematical drawings are created in Geogebra.