Homogeneous system of linear equation:

In summary, the conversation is about solving a homogeneous system of linear equations with three variables and two equations. The solution involves choosing one variable as a parameter and solving for the other two variables in terms of the parameter. The possible answers provided have some errors and the conversation also includes a brief tangent about multiple choice questions in mathematics.
  • #1
TonyC
86
0
I am having trouble finding the solution to the homogeneous system of linear equations:
2x-2y+z=0
-2x+y+z=0
 
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  • #2
I guess I should have also put:

How can I break this down?
 
  • #3
Perhaps I have been out of the rigid math stuff for a while, but how are you going to solve a system with three unknowns and only two equations?
 
  • #4
Choose one variable (e.g. z) as "t" and solve the system for x and y in function of t.
You'll get an infinite number of solutions, for every t (so z), you have a couple (x,y).
 
  • #5
Hence my problem..
The answers to choose from are:
x=3/4t, y=-t,z=1/2t
x=-3/4t,y=-t,z=1/2t
z=-3/4t,y=t,z=1/2t
z=3/4t,y=t,z=1/2t

This is why I am stumped.
 
  • #6
In your case, y was substituted for t. Then solve it as if t was a parameter for x and z.
 
  • #7
? I am still confused.
 
  • #8
You start with

[tex]\left\{ \begin{gathered}
2x - 2y + z = 0 \hfill \\
- 2x + y + z = 0 \hfill \\
\end{gathered} \right[/tex]

Substitute y = t, t is now a parameter, and solve the following (2x2)-system for x and z

[tex]\left\{ \begin{gathered}
2x + z = 2t \hfill \\
- 2x + z = - t \hfill \\
\end{gathered} \right[/tex]
 
  • #9
You might also want to review your list of possible answers. "x" got changed to "z" in some of them!

(Am I the only person who hates multiple choice questions in mathematics?)
 
  • #10
HallsofIvy said:
(Am I the only person who hates multiple choice questions in mathematics?)
(No :yuck:)
 

What is a homogeneous system of linear equations?

A homogeneous system of linear equations is a set of equations in which all the constant terms are equal to zero. In other words, the right-hand side of each equation is equal to zero.

What is the difference between a homogeneous and non-homogeneous system of linear equations?

The main difference between a homogeneous and non-homogeneous system of linear equations is that in a homogeneous system, all the constant terms are equal to zero, while in a non-homogeneous system, at least one of the constant terms is non-zero.

How do you solve a homogeneous system of linear equations?

To solve a homogeneous system of linear equations, you can use methods such as substitution, elimination, or Gaussian elimination. These methods involve manipulating the equations to find the values of the variables that satisfy all the equations at the same time.

What are the possible solutions to a homogeneous system of linear equations?

A homogeneous system of linear equations can have either a unique solution, infinitely many solutions, or no solution. This depends on the number of equations and variables in the system.

Why are homogeneous systems of linear equations important in mathematics and science?

Homogeneous systems of linear equations are important in mathematics and science because they represent systems that have a balance or symmetry. They are also used to model real-life situations and can be solved to find solutions to problems in various fields such as economics, engineering, and physics.

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