An ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.
The same two numbers in a different order give a different point. Always move across by the first number (x), then up or down by the second (y).
Formal definition and notation
An ordered pair (a, b) is a pair of numbers in a fixed order, so (a, b) = (c, d) only when a = c and b = d. On the coordinate plane the first number is the x-coordinate, the horizontal distance from the origin (positive to the right), and the second is the y-coordinate, the vertical distance (positive up). The point (0, 0) is the origin. The graph of a function is the set of ordered pairs (input, output).
The axes split the plane into four quadrants: I (+, +), II (−, +), III (−, −) and IV (+, −), numbered counterclockwise from the upper right.
A point on the x-axis has y = 0, like (5, 0). A point on the y-axis has x = 0, like (0, −3).
Each row of an input-output table is an ordered pair (input, output), such as (2, 7) for y = 3x + 1.
Ordered pairs can record any two linked quantities, such as (hours worked, dollars earned).
Two points with the same y-coordinate lie on a horizontal line, and their distance apart is the difference of their x-coordinates.
How to plot (x, y)
How to plot (x, y)
start at (0, 0)→ move x across → move y up or down
x
positive: go right; negative: go left; 0: stay on the y-axis
y
positive: go up; negative: go down; 0: stay on the x-axis
Alphabetical order helps: x comes before y, and you walk across the floor (x) before you climb the stairs (y).
Try it: drag the points
Interactive
Drag point A or B, or select one and use the arrow keys. The readout gives each ordered pair, the moves from the origin and the quadrant. Try putting A and B at (4, 1) and (1, 4).
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Ordered Pair examples, step by step
Example 1: Which quadrant?
Warm-up
Plot (−3, 5). In which quadrant is it?
Start at the origin.
The x-coordinate is −3: move 3 units left.
The y-coordinate is 5: move 5 units up.
Left and up is the upper-left region, Quadrant II, where points have the signs (−, +).
Answer: The point (−3, 5) is in Quadrant II.
Example 2: Write the pair from a description
Core
Point P is 6 units to the left of the y-axis and 2 units below the x-axis. Write its ordered pair.
Left of the y-axis means a negative x-coordinate: x = −6.
Below the x-axis means a negative y-coordinate: y = −2.
Write x first: (−6, −2). Both negative means Quadrant III.
Answer: P is the point (-6, -2), in Quadrant III.
Example 3: Ordered pairs on a line
Test-style
Which of the ordered pairs (2, 5), (5, 2), (−1, −4) and (0, 1) lie on the line y = 3x − 1?
Substitute each x-coordinate and compare with the y-coordinate.
(2, 5): 3(2) − 1 = 5. Yes.
(5, 2): 3(5) − 1 = 14 ≠ 2. No: this is (2, 5) with the numbers swapped.
A ratio such as 2 : 5 compares two quantities. An ordered pair (2, 5) gives a position, and (4, 10) is a different point even though the ratio is the same.
Quick check: 3 questions
Question 1 of 3
What is the x-coordinate of (−7, 3)?
Show answer and explanation
Answer: A. The x-coordinate is the first number, −7. 3 is the y-coordinate, 7 drops the sign, and −4 is the sum of the two coordinates.
Question 2 of 3
Which point is in Quadrant IV?
Show answer and explanation
Answer: B. Quadrant IV has a positive x and a negative y: (2, −5). (−2, −5) is in III, (−2, 5) is in II, and (5, 0) is on the x-axis, in no quadrant.
Question 3 of 3
Point A is at (1, 2). Point B is 3 units right of A and 4 units down. What are the coordinates of B?
Show answer and explanation
Answer: D. Right 3 adds 3 to x: 1 + 3 = 4. Down 4 subtracts 4 from y: 2 − 4 = −2. So B is (4, −2). Choice A moves up instead of down, B moves left, and C uses each move on the wrong coordinate.
Ordered Pair: frequently asked questions
Why is it called an ordered pair?
Because the order of the two numbers is part of the information. A pair like {2, 5} as a set would be the same as {5, 2}, but the ordered pair (2, 5) is not the same as (5, 2): one is 2 right and 5 up, the other 5 right and 2 up. The first slot is always x and the second is always y.
Can the numbers in an ordered pair be fractions or decimals?
Yes. Any real numbers can be coordinates. The point (2.5, −12) is halfway between 2 and 3 on the x-axis and half a unit below it. Grids drawn with whole-number lines just make whole-number points easier to read; the points in between are still there.
What is the origin?
The origin is the point (0, 0), where the x-axis and the y-axis cross. Every ordered pair tells you how to get to a point starting from the origin: the first number gives the move across and the second gives the move up or down. The origin is not in any quadrant.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). 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Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 12 are not integers.Number System · Grade 6, Grade 7, Grade 8PointA point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.Geometry · Grade 4, Grade 5, Grade 6SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1InputAn input is the value you put into a function or a rule. The rule acts on the input to produce the output: for f(x) = x + 2, the input 5 gives the output 7. For many functions, the input is the x-value. 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