Earlier in the unit we briefly mentioned the idea of indeterminate equations - equations in which the answer cannot be determined. It was also mentioned that there is such a thing as an indeterminate system. There is also the possibility of an inconsistent system. In this section we'll explore both of these ideas.

Indeterminate Systems

An indeterminate system is a system of equations in which it's not possible to determine values for the variables. Here's an example:

x + 3y = 10
2x = -6y + 20

We could use either the addition method or the substitution method to solve this. I'm going to use the substitution method, because I realize that if I divide the second equation by 2, it'll be solved for x:

x = -3y + 10

Now I plug that into the first equation:

x + 3y = 10
(-3y + 10) + 3y = 10

Combining like terms on the left-hand side gives me the following very interesting equation:

10 = 10

Uh...yeah. That's true. But...it doesn't have any variables in it! How can I solve for x and y if I have an equation that doesn't have either?

Well, it turns out that if you use the addition method or the substitution method on a system of equations, and you end up with an obviously true equation (like 10 = 10, or 5 = 5, or 0 = 0) that doesn't have any variables in it, that means the system is indeterminate. It can't be solved because there is an infinite number of solutions.

Remember how, in the first section, we made lists of pairs of numbers that solved the first equation, and then plugged them into the second equation to see which one worked? Well, in an indeterminate system, any pair that works in the first equation will also work in the second equation!

Inconsistent Systems

Indeterminate systems have an infinite number of solutions. Inconsistent systems have no solutions at all. There is no pair of numbers which works as a solution to both equations. Here's an example:

x + 2y = 17
2x + 4y = 32

I see that if I multiply the first equation by -2, and then add them, the x variable will cancel out:

-2x - 4y = -34
 2x + 4y =  32
______________
       0 = -2

What? Zero equals negative two? Who ever heard of such a ridiculous thing? Of COURSE zero does not equal negative two! This is similar to an indeterminate system, except that, instead of having an obviously TRUE equation, we have an obviously FALSE equation. When you get something absurd like this, it's an inconsistent system. There are no values of x and y that make both of the equations true.

When solving a system of equations, if the system is inconsistent, you can write "inconsistent" or "no solution." If it is indeterminate, simply write "indeterminate."