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Coordinates — Definition, Examples & Graph

Coordinates
Ordered Pair
Ordered Triple
n-tuple

On the coordinate plane, the pair of numbers giving the location of a point (ordered pair). In three-dimensional coordinates, the triple of numbers giving the location of a point (ordered triple). In n-dimensional space, a sequence of n numbers written in parentheses.

 

Ordered pair: Two numbers written in the form (x, y).
Ordered triple: Three numbers written in the form (x, y, z).
n-tuple: n numbers written in the form (x1, x2, x3, . . . , xn).

 

 

See also

Analytic geometry

Key Formula

P=(x,y)P = (x, y)
Where:
  • PP = A point on the coordinate plane
  • xx = The horizontal distance from the origin (the x-coordinate or abscissa)
  • yy = The vertical distance from the origin (the y-coordinate or ordinate)

Worked Example

Problem: A treasure map marks a spot 3 units to the right of and 5 units above the origin. What are the coordinates of the treasure?
Step 1: Identify the horizontal position. The point is 3 units to the right, so the x-coordinate is 3.
x=3x = 3
Step 2: Identify the vertical position. The point is 5 units up, so the y-coordinate is 5.
y=5y = 5
Step 3: Write the coordinates as an ordered pair, with the x-coordinate first and the y-coordinate second.
P=(3,5)P = (3, 5)
Answer: The coordinates of the treasure are (3, 5).

Another Example

Problem: Plot the point A = (−2, 4) and determine which quadrant it lies in.
Step 1: Read the x-coordinate: −2. Starting at the origin, move 2 units to the left.
x=−2x = -2
Step 2: Read the y-coordinate: 4. From that position, move 4 units up.
y=4y = 4
Step 3: Since the x-coordinate is negative and the y-coordinate is positive, the point falls in Quadrant II.
Answer: Point A = (−2, 4) is located in Quadrant II of the coordinate plane.

Frequently Asked Questions

Which coordinate comes first, x or y?
The x-coordinate (horizontal position) always comes first, and the y-coordinate (vertical position) comes second. Think of it alphabetically: x before y. This order matters — the point (3, 5) is different from the point (5, 3).
What is the difference between coordinates and an ordered pair?
An ordered pair is the specific notation (x, y) used to write coordinates in two dimensions. The term "coordinates" is more general and can refer to two numbers, three numbers (x, y, z), or even more in higher-dimensional spaces. Every ordered pair is a set of coordinates, but coordinates can also be ordered triples or n-tuples.

Coordinates vs. Ordered pair

An ordered pair refers specifically to two numbers written as (x, y). Coordinates is the broader term that includes ordered pairs in 2D, ordered triples (x, y, z) in 3D, and n-tuples in higher dimensions.

Why It Matters

Coordinates form the foundation of analytic geometry, connecting algebra and geometry by letting you describe geometric shapes with equations. Whenever you graph a line, calculate a distance, or find a midpoint, you rely on coordinates. Beyond math class, coordinates power GPS navigation, computer graphics, and mapping systems worldwide.

Common Mistakes

Mistake: Reversing the order and writing (y, x) instead of (x, y).
Correction: Always write the horizontal value first and the vertical value second. The convention is (x, y), matching alphabetical order. Swapping them places the point in a completely different location.
Mistake: Confusing the sign of coordinates when a point is to the left of or below the origin.
Correction: Points to the left of the origin have a negative x-coordinate, and points below the origin have a negative y-coordinate. Always check direction relative to the origin before assigning signs.

Related Terms