## Independent and Dependent Systems

An independent system of equations is a system with a unique solution. All the examples and exercises in 7.2 were examples of independent systems.

Consider the following system:

$$\mathbb{S}_1$$

$$\begin{eqnarray*} x + y + z &=& 1\\ y + z &=& 2 \end{eqnarray*}$$

Since there are fewer equations in the system than variables, there will not be a unique solution. In order to characterize the solutions of this system, we add a third equation to represent all possible values of $$z$$:

$$\mathbb{S}_2$$

$$\begin{eqnarray*} x + y + z &=& 1\\ y + z &=& 2\\ z &=& t \end{eqnarray*}$$

where $$t$$ can be any real number. Using back substitution, we find

$$\mathbb{S}_3$$

$$\begin{eqnarray*} x + y + z &=& 1\\ y\quad &=& 2 - t\\ z &=& t \end{eqnarray*}$$

and

$$\mathbb{S}_4$$

$$\begin{eqnarray*} x \quad\quad &=& -1\\ y\quad &=& 2 - t\\ z &=& t \end{eqnarray*}$$

So the solution is the set of all ordered triplets of the form $$( - 1, 2 - t, t )$$ where $$t$$ is a real number.

This is an example of a dependent system.

Dependent systems have an infinite number of solutions, since there is a different solution for each value of $$t$$.

## Exercise 7.3.1

Solve the system

$$\mathbb{S}_1$$

$$\begin{eqnarray*} x + 2 y + z &=& 0\\ y - z &=& 2\\ \end{eqnarray*}$$

See Solution

## Inconsistent Systems

Consider the system

$$\mathbb{S}_1$$

$$\begin{eqnarray*} x + y + z &=& 1\\ y + z &=& 2\\ y + z &=& 3 \end{eqnarray*}$$

If we attempt to triangularize the system by $$E_3 \rightarrow - E_2 + E_3$$ we get the system

$$\mathbb{S}_2$$

$$\begin{eqnarray*} x + y + z &=& 1\\ y + z &=& 2\\ 0 &=& 1 \end{eqnarray*}$$

Since the last equation is false, this system has no solution.

The system is said to be inconsistent.

## Exercise 7.3.2

Solve the system

$$\mathbb{S}_2$$

$$\begin{eqnarray*} x - y + z &=& 1\\ x\quad\,\,\,\, + z &=& 1\\ x + y + z &=& 2 \end{eqnarray*}$$

See Solution