- #1
Panphobia
- 435
- 13
Homework Statement
|1 0-1-2-8 | -3|
|0 0 1 2 7 | 1|
|0 0 0 1 -4 | -13|
I think that x[itex]_{2}[/itex] is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?
You could say that x2 is arbitrary or you could say that x2 = t, an arbitrary real number.Panphobia said:Homework Statement
|1 0-1-2-8 | -3|
|0 0 1 2 7 | 1|
|0 0 0 1 -4 | -13|
I think that x[itex]_{2}[/itex] is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?
Panphobia said:Homework Statement
|1 0-1-2-8 | -3|
|0 0 1 2 7 | 1|
|0 0 0 1 -4 | -13|
I think that x[itex]_{2}[/itex] is a free variable, so when writing the general solution in a solution set, what would I put for it? Nothing?
Ray Vickson said:When you write out the equations in detail you see that there is no ##x_2## anywhere in the system. I would not put anything for it; it does not "exist" for this system, no more than ##x_{17}## or ##x_{265}## exist here. However, I suppose you *could* argue the point.
Dick said:I would say ##x^2+y^2=1## in ##R^3## represents a cylinder. Saying it's a circle in the plane because z doesn't occur isn't really a good answer.
To find the solution to a system of linear equations with a free variable, you first need to solve for the other variables in terms of the free variable. Then, you can assign a value to the free variable and solve for the remaining variables.
A free variable in a system of linear equations is a variable that can take on any value. It is not restricted by any of the equations in the system and can be assigned a value to find a solution.
A system of linear equations will have a free variable if there are fewer equations than variables. This means that there are not enough equations to fully determine the values of all the variables, and one variable will be left free to take on any value.
Yes, a system of linear equations can have more than one free variable. This occurs when there are more variables than equations in the system. Each extra variable will be considered a free variable and can take on any value.
Solving systems of linear equations with a free variable can be used in many real-life applications, such as solving optimization problems and modeling real-world situations. For example, it can be used to find the best combination of ingredients for a recipe or to determine the most cost-effective way to produce a product.