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7.3 Independent, Dependent and Inconsistent Systems

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