The ordered pair calculator is an online tool that displays the ordered pair for every equation. The online ordered pairs calculator tool at STUDYQUERIES makes the calculation faster and displays the ordered pairs in a fraction of a second.
How to Use the Ordered Pairs Calculator?
To use the ordered pairs calculator, follow these steps:
- Step 1: Enter the equation in the input box
- Step 2: Click the button “Submit” to retrieve the ordered pairs
- Step 3: The ordered pair for the equation will be displayed in the output box
What is an Ordered Pair?
According to its name, an ordered pair is a pair of elements whose placement has special significance. In coordinate geometry, ordered pairs are typically used to represent points on a coordinate plane. They can also be used to represent elements of a relation.
Discover more about ordered pairs, including their definition, meaning, properties, and more.
An ordered pair is a pair formed by two elements that are separated by a comma and written inside the parentheses. For example, \((x, y)\) represents an ordered pair, where ‘x’ is called the first element and ‘y’ is called the second element of the ordered pair.
The names of these elements vary depending on what context they are used in and they can either be variables or constants. The order of the elements in an ordered pair is important. It means \((x, y)\) may not be equal to \((y, x)\) all the time.
\((2, 5)\), \((a, b)\), \((0, -5)\), etc. are examples of ordered pairs.
Ordered Pair in Coordinate Geometry
In coordinate geometry, an ordered pair represents the position of a point on the coordinate plane with respect to the origin. A coordinate plane is formed by two perpendicular intersecting lines among which one is horizontal \((x-axis)\) and the other line is vertical \((y-axis)\).
The origin is the intersection of the two axes. Every point on the coordinate plane is represented by an ordered pair \((x, y)\) where the first element \(x\) is called the \(x-coordinate\) and the second element \(y\) is called the \(y-coordinate\). It is possible to see more differences between the elements of the ordered pair used in geometry here.
First Element of Ordered Pair
- It is called \(x-coordinate\).
- Another name for this is \(“abscissa”\).
- It represents the horizontal distance from the origin of the point.
- This number is one of the numbers on the \(x-axis\).
- It represents the distance of the point from the \(y-axis\).
Example: If \((2, 4)\) is a point on the coordinate plane, then \(2\) is the distance of the point from the \(y-axis\).
Second Element of Ordered Pair
- It is called \(y-coordinate\).
- Another name for this is \(“ordinate”\).
- A vertical distance is a distance that a point is from the origin.
- This number is one of the numbers on the \(y-axis\).
- It represents the distance of the point from the \(x-axis\).
Example: If \((2, 4)\) is a point on the coordinate plane, then \(2\) is the distance of the point from the \(y-axis\).
Graphing Ordered Pairs
Now, we understood the difference between the \(x-coordinate\) and \(y-coordinate\) of an ordered pair in coordinate geometry. In the next section, we will see how ordered pairs are graphed.
- Step 1: Always start from the origin and move horizontally by \(|x|\) units to the right if \(x\) is positive and to the left, if \(x\) is negative. Stay there.
- Step 2: Start from where you have stopped in Step 1 and move vertically by \(|y|\) units to up if \(y\) is positive and to down if \(y\) is negative. Stay there.
- Step 3: Place a dot exactly at the point where you have stopped in Step 2 and that dot represents the ordered pair \((x, y)\)
In these steps, \(|x|\) and \(|y|\) represent the absolute values of \(x\) and \(y\) respectively.
Example: Graph the ordered pair \((4, -3)\).
Let’s start from the origin, move to the right by 4 units \((as\ 4\ is\ positive)\) and then move down by 3 units \((as\ 3\ is\ negative)\).
The order of the elements in an ordered pair is crucial, hence the name “ordered pair”. For example, \((4, -3)\) and \((-3, 4)\) are located at different positions on the plane as shown below.
Ordered Pairs in Different Quadrants
In the above figure, we can see that the coordinate plane is divided into four parts by the x and y axes. These four parts are called quadrants. The signs of x and y in an ordered pair \((x, y)\) of a point differs depending upon the quadrant and they are shown in the table below.
Quadrant Ordered Pair Signs
- \(x > 0, y > 0\)
- \(x < 0, y > 0\)
- \(x < 0, y < 0\)
- \(x > 0, y < 0\)
For example, \((2, -4)\) refers to 2 on the x-axis \((positive)\) and -4 on the y-axis \((negative)\). So \((2, -4)\) is a point in quadrant IV.
Ordered Pair in Sets
We have seen that ordered pairs are used in coordinate geometry to locate a point. They are also used in set theory but in a different way. In mathematics, the cartesian product is the set of possible ordered pairs from one set to another.
For example, if \(A = {1, 2, 3}\) and \(B = {a, b, c}\), then the cartesian product is \(A \times B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c), (3, a), (3, b), (3, c)}\) and it is a set formed by all ordered pairs \((x, y)\) where x is in A and y is in B. Any subset of the cartesian product is called a relation. For example, \({(1, a), (1, b), (3, c)}\) is a relation.
Examples:
If \((2, 4)\) is part of the relation “divides”, it means that 2 divides 4.
If \((4, 2)\) is in the relation “greater than”, that means 4 is greater than 2.
If \((x, y)\) is part of a relation “is a sister of”, it means that x is a sister of y.
Equality Property of Ordered Pairs
For any two ordered pairs \((x, y)\) and \((a, b)\) \((either\ in\ coordinate\ geometry\ or\ in\ relations)\), if \((x, y)\) = \((a, b)\) then x = a and y = b. If two ordered pairs are equal, then their corresponding elements are also equal. This is known as the equality property of ordered pairs. As an example:
If \((x, y)\) = \((2, -3)\) then \(x = 2\) and \(y = -3\).
If \((x + 1, y – 2)\) = \((-3, 5)\) then \(x + 1 = -3\) and \(y – 2 = 5\).
Important Points on Ordered Pairs
- An ordered pair \((x, y)\) represents the location of a point in coordinate geometry where x is the horizontal distance and y is the vertical distance.
- An ordered pair \((x, y)\) represents an element of a relation R which is denoted by \(xRy\) \((x “is\ related\ to” y)\).
- If \((x, y)\) = \((a, b)\) then \(x = a\) and \(y = b\).
FAQs
What is an example of an ordered pair?
An ordered pair is a pair of numbers in a specific order. For example, \((1, 2)\) and \((- 4, 12)\) are ordered pairs. The order of the two numbers is important: \((1, 2)\) is not equivalent to \((2, 1)\).
What is the ordered pair form?
An ordered pair contains the coordinates of one point in the coordinate system. A point is named by its ordered pair of the form of \((x, y)\). In this example, the first number corresponds to the x-coordinate and the second to the y-coordinate. A point is graphed by drawing a dot at the coordinates that correspond to the ordered pair.
What is an ordered and unordered pair?
In mathematics, an unordered pair or pair set is a set of the form \({a, b}\), i.e. a set having two elements \(a\) and \(b\) with no particular relation between them, where \({a, b} = {b, a}\). In contrast, an ordered pair \((a, b)\) has an as its first element and b as its second element, which means \((a, b) \neq (b, a)\).
What is an ordered pair in relation and function?
An ordered pair is represented as (INPUT, OUTPUT): This shows the relationship between INPUT and OUTPUT. In contrast, a function is a relationship that derives one OUTPUT for each INPUT. Note that not all relations are functions.
Is the ordered pair a solution?
Ordered pairs \((x,y)\) that work in both equations are called solutions to the system of equations. Such pairs indicate where two lines intersect. The solution to a system can be one, zero, or infinitely many.