Quick question about free variables

[wumple]

Hopefully I have this in the right place, it's not a homework question exactly, rather a question I have as I'm reading through my text. I'm learning about free variables. The book gives the example system:

x - z = 2
y + 2z = -1
0 = 0

as an example of a system with infinitely many solutions. I see that the way to express the solution set for this system is by describing the line that the solutions lie on. It says to make z the 'free variable' and make x and y the 'dependent' variables. Then by picking z I can find values for x and y that work. This makes sense to me. But why can't I solve for, say, x and z in terms of y? Why do I have to pick z as the free variable?

 

[antibrane]

The free parameter you choose is arbitrary, so just as you could say, like the book, that z is free and then

$$(x,y,z)=(2+z,-1-2z,z)$$

you could also have said that x was your free parameter and stated the solutions as

$$(x,y,z)=(x,3-2x,-2+x)$$

I think they must have chosen z because, in this case, it is easier to solve for x and y.

 

[wumple]

Thanks so much!

 

[antibrane]

Anytime!

 

[blue_raver22]

I would say that the reason for choosing z as the free variable is due to the structure of the equations in the system. In this particular example, there are two equations and three unknown variables. This means that we have one less equation than unknowns, which leads to an infinite number of solutions. In order to express these solutions, we need to choose one variable as the free variable and express the other two in terms of it.

Now, why did we choose z as the free variable instead of x or y? This is because the equations are written in a way that makes it easier to solve for z. In the first equation, z is already isolated on one side, while in the second equation, z appears with a coefficient of 2, making it easier to solve for. This is not always the case, and sometimes it may be more convenient to choose a different variable as the free variable.

In terms of solving for x and z in terms of y, it is certainly possible to do so, but it would result in a different form of the solution set. In this case, we would have to express x and z in terms of y and then describe the solution set as a plane in 3D space, rather than a line. It all depends on how we choose to express the solutions and what is most convenient for our purposes.

Overall, the concept of free variables is used to express the infinite number of solutions in a concise and understandable way, and the choice of which variable to make free depends on the structure of the equations in the system. I hope this clarifies your understanding of free variables.