how would I get the solution to x y and z

Question

Solve the system.

−2x+12y+2z=−21

 

−x+y−4z=8

 

1/3x+2y+4z=-18

Answer 1

You're almost there.  Given those 3 math statements, we can now do a matrix.  It would look like this:

 

-2          12         2          =          21

-1            1       -4          =            8

1/3          2         4          =         -18

 

First thing is to get rid of that fraction in the 3rd row, by doing the following row operation:

 

3R3 = new R3; which gives us the new matrix of:

 

-2          12          2          =          21

-1            1         -4         =             8

  1            6        12         =         -54

 

I prefer to use a system that gives great certainty of not having a 0 show up where one has already been placed; which leads to a very efficiently solved matrix.  So, do the following row operation:

 

[1]   R2 + R3 = new R3 ; which gives us the new matrix of:

 

-2          12          2          =          21

-1            1          4          =            8

 0            7          8          =         -46

 

Then, I want to get the value in R2, C1 equal to 0; so, I do the following row operation:

 

[2]   R1 - 2R2 = new R2; which gives us the new matrix of:

 

-2          12         2          =          21

 0           10       -6          =            5

 0             7        8          =         -46

 

Then, I want to get the value in R3, C2 equal to 0; so I do the following row operation:

 

[3]  7R2 - 10R3 = new R3; which gives us the matrix of:

 

-2          12        2          =          21

 0           10       -6         =            5

 0             0       38         =       -425

 

Continue in this same type of pattern, by getting 0's to:

[4]  R1, C3

[5]  R2, C3

[6]  R1, C2

 

Follow this up by row reductions.  Then, you have your x, y, and z values

 

 

Answer 2

Assuming that the student is in Algebra II or the equivalent, the student can use various methods taught at that level:

1. On calculator, using Reduced Row Echelon function RREF (a MATRIX method)

2. Determinants / Cramer's rule

3. Matrix Inverse solution to AX = B

4. Substitution/elimination method, by hand, usually the least attractive choice.