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Elementary row operations
Linear Algebra. Inner Product, Norm, and Orthogonal Vectors. Ring theory. Group Theory. Problem 24 Solve the following system of linear equations using Gaussian elimination.
Related Question. Elementary row operations. Determine all the […] Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations or matrix equations described below has no solution, one unique solution or infinitely many solutions and justify your answer. Warning: Since the third equation has an x -term, I cannot use it on either of the other two equations any more or I'll undo my progress.
Finite Math Examples
I can work on the equation, but not with it. If I add twice the first row to the second row, this will give me a leading 1 in the second row. I won't have gotten rid of the leading y -term in the second row, but I will have converted it without getting involved in fractions to a form that is simpler to deal with. You should keep an eye out for this sort of simplification.
- The Elimination Method
First I do the scratch work:. Now I can use the second row to clear out the y -term in the first row.
I can tell what z is now, but, just to be thorough, I'll divide the first row by Then I'll rearrange the rows to put them in upper-triangular form:. Now I can start the process of back-solving:. Note: There is nothing sacred about the steps I used in solving the above system; there was nothing special about how I solved this system. You could work in a different order or simplify different rows, and still come up with the correct answer. These systems are sufficiently complicated that there is unlikely to be one right way of computing the answer. So don't stress over "how did she know to do that next?
I just did whatever struck my fancy; I did whatever seemed simplest or whatever came to mind first. Don't worry if you would have used completely different steps.
As long as each step along the way is correct, you'll come up with the same answer. In the above example, I could have gone further in my computations and been more thorough-going in my row operations, clearing out all the y -terms other than that in the second row and all the z -terms other than that in the first row. This is what the process would then have looked like:.
This way, I can just read off the values of x , y , and z , and I don't have to bother with the back-substitution. Many texts only go as far as Gaussian elimination, but I've always found it easier to continue on and do Gauss-Jordan.
Note that I did two row operations at once in that last step before switching the rows. As long as I'm not working with and working on the same row in the same step, this is okay. In this case, I was working with the first row and working on the second and third rows. Stapel, Elizabeth. Accessed [Date] [Month] Linking to this site School licensing.
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