Examples of ordered pair in the following topics:
-
- 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.
-
- In an equation where $x$ is a real number, the graph is the collection of all ordered pairs with any value of $y$ paired with that real number for $x$.
- For example, to graph the equation $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
-
- Allowing $y=f(x)$, where $f(x)=|x|$, some ordered pair examples of $(x,|x|)$ 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), $f(x)=3$ (a constant function) for the domain $1ordered pairs are:
- For the last part (piece), $f(x)=x$ for the domain $x>2$, a few ordered pairs are:
-
- The ordered pairs normally stated in linear equations as $(x,y)$, in function notation are now written as $(x,f(x))$.
- Next, substitute these values into the function for $x$, and solve for $f(x)$ (which means the same as the dependent variable $y$): we get the ordered pairs:
- Next, plug these values into the function, $f(x)=x^{3}-9x$, to get a set of ordered pairs, in this case we get the set of ordered pairs:
-
- We can also tell by the set of ordered pairs given in this mapping that it is a function because none of the $x$-values repeat: $(-1,1),(1,1),(7,49),(0.5,0.25)$; 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.
-
- 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 $(x, y)$.
-
- A list of ordered pairs for the function are:
- The ordered pairs $(-2,4)$ and $(2,4)$ do not pass the definition of one-to-one because the element $4$ of the range corresponds to to $-2$ and $2$.
- 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.
-
- Each solution is an ordered pair and can be written in the form $(x, y)$.
- There are thus an infinite number of ordered pairs that satisfy the equation.$$
- Note that the ordered pair $(3, 10)$ tells us that $x = 3$ and $y = 10$.
-
- After creating a few $x$ and $y$ 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 $(10,0)$ and $(-10,0)$.
- The line continues on to infinity in each direction, since there is an infinite series of ordered pairs of solutions.
-
- 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.