In of Problem, use the method of elimination to determine whether the given linear system...

In of Problem, use the method of elimination to determine whether the given linear system is consistent or inconsistent. For each consistent system, find the solution if it is unique; otherwise, describe the infinite solution set in terms of an arbitrary parameter t (as in Examples 5 and 7).

x – 4y = −10

-2x + 4y = 20

Example 5

To solve the system

we reduce its augmented coefficient matrix to echelon form as follows.

Our final result is the echelon matrix in (16), so by Eqs. (17a)-(17d) in Example 4, the infinite solution set of the system in (18) is described in terms of arbitrary parameters s and t as follows:

Thus, the substitution of any two specific values for s and t in (19) yields a particular solution (x1, x2, x3, x4,..x5) of the system, and each of the system’s infinitely many different solutions is the result of some such substitution.

Examples 3 and 5 illustrate the ways in which Gaussian elimination can result in either a unique solution or infinitely many solutions. On the other hand, if the reduction of the augmented matrix to echelon form leads to a row of the form

0 0 … 0 0 *

where the asterisk denotes a nonzero entry in the last column, then we have an inconsistent equation,

0x1 + 0x2 + … 0xn = *

and therefore the system has no solution.

Remark: We use algorithms such as the back substitution and Gaussian elimination algorithms of this section to outline the basic computational procedures of linear algebra. In modern numerical work, these procedures often are implemented on a computer. For instance, linear systems of more than four equations are usually solved in practice by using a computer to carry out the process of Gaussian elimination.