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Definitions

Consistent

A consistent system of equations is one that has at least one solution. If you have the system:

$$\begin{cases} x + y = 10\\ 2x + 2y = 20\\ \end{cases}$$ That's consistent, because the solutions are the line $x + y = 10$.

Inconsistent

If a system of equations has no solutions, then it is inconsistent. If the last column (in an augmented matrix) is a pivot column, that is, it has a pivot, then it's inconsistent.

$$\begin{cases} x + y = 10\\ 2x + 2y = 21\\ \end{cases}$$ That's inconsistent, if you subtract the second equation from 2 times the first, you get $(2x + 2y = 21) - 2(x + y = 10) \rightarrow 0 = 1$. That makes the system false, so there are no solutions to that system of equations.

Basic and Free Variables

A basic variable is one that is bound by an equation. A free variable is not bound by any equation. Here is an example:

$$\begin{bmatrix}3x_1 & 4x_2 & & 2x_4 & 1\\ & x_2 & & 2x_4 & 8\\ 6x_1 & & x_3 & & 10 \end{bmatrix} \to$$ $$\begin{bmatrix}x_1 & & & -2x_4 & \frac{-31}{3}\\ & x_2 & & 2x_4 & 8\\ & & x_3 & 14x_4& 72 \end{bmatrix}$$ $x_1$ is a basic variable. It is bound by the equation $2x_4 - \frac{31}{3}$ So is $x_2$. It equals $8 - 2x_4$. So is $x_3$. It equals $72 - 14x_4$. $x_4$ is a free variable. It isn't bound by any equation.

So, to determine whether a variable is basic or free, check whether it has a pivot value. In the matrix above, the first, second, and third columns were pivot columns, meaning those three variables were basic, while the fourth was free. If the fifth column, or the augmented column, is a pivot column, it's inconsistent, so there is no solution at all.

Determine Whether a System of Equations is Consistent

This is very simple. This requires two steps.

1. Convert to Row-Eschilon Form

2. Check if the last column is a pivot column

1. If it is, it's inconsistent

2. If it isn't, it's consistent

Brief Explanation

If the last column is a pivot column, then that row gives an equation that looks something like 0x + 0y + 0z = 1, meaning 0 = 1. Clearly, this is false, so the system of equations is inconsistent.