Solve the system of equations by finding the reduced row-echelon form of the augmented matrix for the system of equations. 2x+y+z=-3 3x-5y+3z=-4 5x-y+2z=-2

  (x, y, z) = (1, -1, -4)

Step-by-step explanation:

A suitable graphing or scientific calculator can find the reduced row-echelon form for you. There are on-line calculators that will do that, too.

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In general, if you want to do this by hand, you want to use row operations on the augmented matrix to make the diagonal elements 1 and the off-diagonal elements 0 as shown in the attached result.

If a[i,j] represents the element at row i, column j, you do that by dividing row i by a[i, i] (to make a[i, i] = 1), then subtracting the product of row i and a[k,i] from row k. (for all rows k ≠ i) For this 3-row matrix, repeat these steps for i = 1 to 3.

In the general case of an n by n+1 augmented matrix, you will be doing n^2 row operations, each one involving evaluation of n+1 expressions. The work rapidly grows with matrix size, so readily justifies use of a calculator.

As with many "elimination" problems, appropriate choice of sequence can reduce the work. The above algorithm always produces the reduced row-echelon form, but may result in messy arithmetic along the way.

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