System of Linear Equations - Homogenous Case

[jenl2881]

I'm in Mathematics for Economics and we are using the text "Mathematical Tools for Economics" by Turkington. The question I have comes from the section on the Homogenous Case Ax=0 concerning Linear Equations. The question (Exercise 2.2 question 2):

The system of equations

x₁ + 2x₂ + 2x₃ - 3x₄ = 0
x₁ - 2x₃ + 13x₄ = 0
3x₁ + 5x₂ + 4x₃= 0

has an infinite number of solutions if a nontrivial solution exists. Why? Show that this is the case and obtain the general solution.

I may be over thinking it but I know that there are more variables than equations so I can set x₄ to 0. I also know after this that I have to perform row operations to reduce matrix down to echelon form (I'm not that great at this part).

Any help would be greatly appreciated. The instructor is going really fast and I'm a little lost.

 

[stapel]

The system of equations

x₁ + 2x₂ + 2x₃ - 3x₄ = 0
x₁ - 2x₃ + 13x₄ = 0
3x₁ + 5x₂ + 4x₃= 0

has an infinite number of solutions if a nontrivial solution exists. Why? Show that this is the case and obtain the general solution.

I may be over thinking it but I know that there are more variables than equations...

Yes.

...so I can set x₄ to 0.

No.

How would the presence of an "extra" variable (being what would given infinitely-many non-trivial solutions) cause the value of that variable to be one (finite trivial) value of zero? :shock:

 

[Ishuda]

...I may be over thinking it but I know that there are more variables than equations so I can set x₄ to 0. ...

Sort of, in a sense; but then you can also set it to 1 or 96 or .001 or .... To answer a question like this, the way to solve the equations is to move one of the variables to the other side [ I assume you would also know how to solve the set of equation if there were non zero 'numbers' on the right hand side]. Thus you get

x₁ + 2x₂ + 2x₃ = 3x₄
x₁ - 2x₃ = -13x₄
3x₁ + 5x₂ + 4x₃= 0

For the moment assume that x4 is known (it could even be zero but leave the designation there). You now have (assuming a non-trivial solution) values of x₁, x₂, and x₃, in terms of x₄. Since x₄ can have an infinite number of values, there are an infinite number of solutions.

 

[HallsofIvy]

I'm in Mathematics for Economics and we are using the text "Mathematical Tools for Economics" by Turkington. The question I have comes from the section on the Homogenous Case Ax=0 concerning Linear Equations. The question (Exercise 2.2 question 2):

The system of equations

x₁ + 2x₂ + 2x₃ - 3x₄ = 0
x₁ - 2x₃ + 13x₄ = 0
3x₁ + 5x₂ + 4x₃= 0

has an infinite number of solutions if a nontrivial solution exists. Why? Show that this is the case and obtain the general solution.

I may be over thinking it but I know that there are more variables than equations so I can set x₄ to 0.

Setting x₄ to 0 would just give three equations in three variables but still homogeneous with all equal to 0 being the only solution. Instead set x₄ equal to a non zero value to get three equations in three variables.

I also know after this that I have to perform row operations to reduce matrix down to echelon form (I'm not that great at this part).

Any help would be greatly appreciated. The instructor is going really fast and I'm a little lost.