General solutions

Suppose that the following REF is obtained.

$\displaystyle \left[\begin{array}{rrrr}
1 & 0 & -5 & 1 \\
0 & 1 & 1 & 4 \\
0 & 0 & 0 & 0
\end{array}\right]
$

The variables $ x_1$ and $ x_2$ corresponding to the pivot columns $ \begin{bmatrix}1  0  0 \end{bmatrix}$ and $ \begin{bmatrix}0  1  0 \end{bmatrix}$ are called basic variables. And the variable $ x_3$ corresponding to the column $ \begin{bmatrix}-5  1  0 \end{bmatrix}$ is called a free variable. Then a general solution can be expressed as

$\displaystyle \left\{\begin{array}{l}
x_1 = 1 + 5 t \\
x_2 = 4 - t \\
x_3 = t
\end{array}\right.
$

where $ t$ is a parameter and corresponds to the free variable $ x_3$.

EXAMPLE 3. Find the general solution of the linear system whose augmented matrix has been reduced to

$\displaystyle \left[\begin{array}{rrrrrr}
1 & 6 & 2 & -5 & -2 & -4 \\
0 & 0 & 2 & -8 & -1 & 3 \\
0 & 0 & 0 & 0 & 1 & 7
\end{array}\right]
$



Department of Mathematics
Last modified: 2005-10-21