Consistent And Dependent Systems Have

Learning Outcomes

Identify inconsistent systems of equations containing three variables
Limited the solution of a system of dependent equations containing 3 variables

Merely every bit with systems of equations in two variables, we may come across an inconsistent organisation of equations in three variables, which means that it does not accept a solution that satisfies all three equations. The equations could correspond three parallel planes, 2 parallel planes and one intersecting aeroplane, or iii planes that intersect the other two but not at the same location. The process of emptying volition result in a false argument, such as [latex]3=7[/latex] or some other contradiction.

Infinitely Many or No Solutions

Systems that have an infinite number of solutions are those which, after elimination, outcome in an expression that is ever true, such as [latex]0=0[/latex]. Graphically, an infinite number of solutions represents a line or coincident plane that serves equally the intersection of three planes in space. The graphic below shows how three planes can intersect to course a line giving the system infinitely many solutions.

Infinitely many solutions.

Systems that have no solution are those that, afterward elimination, result in a statement that is a contradiction, such equally [latex]three=0[/latex]. Graphically, a system with no solution is represented by three planes with no point in common. Iii parallel planes (c), ii parallel planes and 1 intersecting aeroplane (b), three planes that intersect the other ii but not at the aforementioned location (a).

In the start case, nosotros will encounter how it is possible to have a organisation with iii variables and no solutions.

Example

Solve the following system.

[latex]\begin{array}{ll}\text{ }x - 3y+z=iv\,\,\,\,\,\,\,\,\,\,\,\,\left(1\right)\\ \,\,\,\,\,\,-y-4z=7\,\,\,\,\,\,\,\,\,\,\,\,\,\left(two\correct)\,\,\,\,\\\,\,\,\,\,\,\,\,2y+8z=-12\,\,\,\,\,\,\,(3)\terminate{assortment}[/latex]

We will bear witness another example of using elimination to solve a system in iii variables that ends upwardly having no solution in the post-obit video.

We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The aforementioned is true for dependent systems of equations in three variables. An space number of solutions can upshot from several situations. The three planes could exist the same so that a solution to i equation volition exist the solution to the other ii equations. All three equations could be different only they intersect on a line, which has infinite solutions. The other possibility is that ii of the equations could exist the same and intersect the third on a line.

Example

Notice the solution to the given system of three equations in three variables.

[latex]\begin{assortment}{rr}\hfill \text{ }2x+y - 3z=0& \hfill \left(1\correct)\\ \hfill 4x+2y - 6z=0& \hfill \left(2\correct)\\ \hfill \text{ }x-y+z=0& \hfill \left(iii\right)\end{array}[/latex]

In our last video example, we prove a organization that has an infinite number of solutions.

Summary

A system with three variables can take one, none, or many solutions.
A system with no solutions will have a not-true result when solving.
A system with many solutions will have an identity outcome when solving.