 Quandaries and Queries

My name is Michael and I'm in the 11th grade.  I have a Math question that I can't solve.  The problem is system of equations that I need to do in augmented matrix form,  find the row echelon form, and solve it by using back substitution.

2x + 3y + 7z =13
3x + 2y - 5z = -22
5x+ 7y - 3z = -28

Please respond. Thank you!

Hi Michael,

There are many ways to arrive at a row echelon form. I try do do as much as I can without introducing fractions.

Multiply the first row by -1 and add it to the second row and then multiply the first row by -2 and add it to the third row. Hence

 2 3 7 13 3 2 -5 -33 5 7 -3 -28

becomes

 2 3 7 13 1 -1 -12 -35 1 1 -17 -54

Now multiply the second row by -1 and add it to the third row and then multiply the second row by -2 and add it to the first row. The matrix then becomes

 0 5 31 83 1 -1 -12 -35 0 2 -5 -19

Next multiply the third row by -2 and add it to the first row to get

 0 1 41 121 1 -1 -12 -35 0 2 -5 -19

Multiply the first row by -2 and add to the third row to get

 0 1 41 121 1 -1 -12 -35 0 0 -87 -261

Finally interchange the first and second rows and multiply the third row by -87 to get the row echilon form

 1 -1 -12 -35 0 1 41 121 0 0 1 3

Thus z = 3. Back substitute to get y and x and then VERIFY that you have the correct solution.

Penny

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