ordered triple
Represent solutions to three-variable systems as ordered triples (x, y, z) in Grade 10 algebra, and verify solutions by substituting into all three original equations.
Key Concepts
The solution to a system of three equations in three variables is an ordered triple and is noted $(x, y, z)$. A solution means that there is an ordered triple that satisfies all three equations.
For the solution $(1, 6, 9)$, the point must satisfy all three equations, like $x+y+z=4$, to be correct. It's the one spot where they all agree! For example, if you test the point $(2, 1, 3)$ in the system $x+y+z=6$, $2x y+z=6$, and $x y z= 2$, it works for all three. So, $(2, 1, 3)$ is the solution.
Think of an ordered triple as the secret coordinates $(x, y, z)$ to a treasure chest! This single point is where three different flat planes, representing your equations, all meet. If your coordinates don't work for all three planes, you've found the wrong spot and there is no treasure for you.
Common Questions
What is an ordered triple?
An ordered triple (x, y, z) is a set of three values that together satisfy a system of three equations in three variables. It represents a point in three-dimensional space.
How do you verify that (1, 2, 3) is a solution to a system?
Substitute x=1, y=2, z=3 into each of the three original equations. If all three equations are satisfied, then (1,2,3) is a valid solution.
What does an ordered triple represent geometrically?
An ordered triple represents a point in 3D coordinate space. When it is the unique solution to a three-variable system, it is the point where three planes all intersect.