ordered pair

(noun)

A set containing exactly two elements in a fixed order, used to represent a point in a Cartesian coordinate system. Notation: (x,y)(x,y)(x,y).

Related Terms

  • quadrant
  • -axis
  • y-axis
  • x-axis
  • substitution method
  • ndependent and Dependent Variables
  • Cartesian coordinates
  • dependent variable
  • independent variable
  • system of equations
  • constraint

Examples of ordered pair in the following topics:

  • Graphs of Linear Inequalities

    • If it is >>> or <<<, then use a dotted or dashed line, since ordered pairs found on the line would result in a false statement.
    • Shading indicates all the ordered pairs in the region that satisfy the inequality.  
    • For example, if the ordered pair is in the shaded region, then that ordered pair makes the inequality a true statement.
    • These overlaps of the shaded regions indicate all solutions (ordered pairs) to the system.
    • All possible solutions are shaded, including the ordered pairs on the line, since the inequality is ≤\leq≤ the line is solid.  
  • Graphs of Equations as Graphs of Solutions

    • In an equation where xxx is a real number, the graph is the collection of all ordered pairs with any value of yyy paired with that real number for xxx.
    • For example, to graph the equation x−1=0,x-1=0, x−1=0, a few of the ordered pairs would include:
    • After substituting the rest of the values, the following ordered pairs are found:
    • After graphing the ordered pairs and connecting the points, we see that the set of (infinite) points follows this pattern:
    • Construct the graph of an equation by finding and plotting ordered-pair solutions
  • Piecewise Functions

    • Allowing y=f(x)y=f(x)y=f(x), where f(x)=∣x∣f(x)=|x|f(x)=∣x∣, some ordered pair examples of (x,∣x∣)(x,|x|)(x,∣x∣) are:
    • Some ordered pair examples are:
    • Those points satisfy the first part of the function and create the following ordered pairs:
    • For the middle part (piece), f(x)=3f(x)=3f(x)=3 (a constant function) for the domain $1ordered pairs are:
    • For the last part (piece), f(x)=xf(x)=xf(x)=x for the domain x>2x>2x>2, a few ordered pairs are:
  • Graphical Representations of Functions

    • The ordered pairs normally stated in linear equations as (x,y)(x,y)(x,y), in function notation are now written as (x,f(x))(x,f(x))(x,f(x)).
    • Next, substitute these values into the function for xxx, and solve for f(x)f(x)f(x) (which means the same as the dependent variable yyy):  we get the ordered pairs:
    • Next, plug these values into the function, f(x)=x3−9xf(x)=x^{3}-9xf(x)=x​3​​−9x, to get a set of ordered pairs, in this case we get the set of ordered pairs:
  • Introduction to Domain and Range

    • We can also tell by the set of ordered pairs given in this mapping that it is a function because none of the xxx-values repeat: (−1,1),(1,1),(7,49),(0.5,0.25)(-1,1),(1,1),(7,49),(0.5,0.25)(−1,1),(1,1),(7,49),(0.5,0.25); since each input maps to exactly one output.  
    • We can also tell this mapping, and set of ordered pairs is a function based on the graph of the ordered pairs because the points do not make a vertical line.  
    • Let's look at this mapping and list of ordered pairs graphed on a Cartesian Plane.
    • In order to find the domain of a function, if it isn't stated to begin with, we need to look at the function definition to determine what values are not allowed.
    • This mapping or set of ordered pairs is a function because the points do not make a vertical line.  
  • Introduction to Systems of Equations

    • In order for a linear system to have a unique solution, there must be at least as many equations as there are variables.
    • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
    • In this example, the ordered pair (4, 7) is the solution to the system of linear equations.
    • We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations.
    • An independent system has exactly one solution pair (x,y)(x, y)(x,y).
  • One-to-One Functions

    • A list of ordered pairs for the function are:
    • The ordered pairs (−2,4)(-2,4)(−2,4) and (2,4)(2,4)(2,4) do not pass the definition of one-to-one because the element 444 of the range corresponds to to −2-2−2 and 222.
    • Notice also, that these two ordered pairs form a horizontal line; which also means that the function is not one-to-one as stated earlier.
  • Equations in Two Variables

    • Each solution is an ordered pair and can be written in the form (x,y)(x, y)(x,y).
    • There are thus an infinite number of ordered pairs that satisfy the equation.
    • Note that the ordered pair (3,10)(3, 10)(3,10) tells us that x=3x = 3x=3 and y=10y = 10y=10.
  • Graphing Equations

    • After creating a few xxx and yyy ordered pairs, we will plot them on the Cartesian plane and connect the points.
    • Through the same arithmetic as above, we get the ordered pairs (10,0)(10,0)(10,0) and (−10,0)(-10,0)(−10,0).
    • The line continues on to infinity in each direction, since there is an infinite series of ordered pairs of solutions.
  • Graphing Inequalities

    • Recall that for a linear equation in two variables, ordered pairs that produce true statements when substituted into the equation are called "solutions" to that equation.
    • We say that an inequality in two variables has a solution when a pair of values has been found such that substituting these values into the inequality results in a true statement.
    • Recall that, in order to graph an equation, we can substitute a value for one variable and solve for the other.
    • The resulting ordered pair will be one solution to the equation.
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