Problem 37 - College Algebra

Question

Describe what happens when Gaussian elimination is used to solve an inconsistent system.

Step-by-Step Solution

Answer

When Gaussian elimination is used to solve an inconsistent system, the process will yield a row where the left side sums to zero and the right side is a non-zero constant, demonstrating that the system doesn't have a solution.

Step 1: Understanding Gaussian Elimination

Gaussian Elimination is a method of solving matrix equations of the form \(Ax = B\). The process involves taking the augmented matrix of the system and using elementary row operations to bring it to a row-echelon or reduced row-echelon form.

Step 2: Recognizing an Inconsistent System

In an inconsistent system, no solution satisfies all the equations simultaneously. When you apply Gaussian elimination method to an inconsistent system, you eventually end up with a row in the form \([0, 0, ..., 0 | C]\) with \(C \neq 0\). This row implies the equation \(0=C\) which is always false since \(C \neq 0\) and proves that the system doesn't have a solution.

Step 3: Conclusion

So, when Gaussian elimination is used to solve an inconsistent system, it results in at least one row where the left side sums to zero and the right side is a non-zero constant, proving the inconsistency of the system.