System of Equations

Question

5x - y + z = 6

3x + 2y - 3z = -19

x - 3y + 2z = 5

Answer 1

5x - y + z = 6

3x + 2y - 3z = -19

x - 3y + 2z = 5

 

{x, y, z} = {2/9, 5, 89/9}

 

I will solve using substitution/elimination methods. However, you may want to look up Gauss-Jordan Elimination and learn this method as well.

 

Step 1

 

Multiply line 3 by 3 and replace.  Thus,

 

3x - 9y + 6z = 15

 

Step 2

 

Subtract the new line 3 from line 2.  Thus,

 

   5x - y + z = 6
   3x + 2y - 3z = -19
-(3x - 9y + 6z = 15)

--------------------------

         11y - 9z = -34 (We need this for the next round of elimination)

 

Step 3

 

Multiply the original line 3 by and replace.  Thus,

 

5x - y + z = 6
3x + 2y - 3z = -19
5x - 15y + 10z = 25

 

Step 4

 

Subtract the new line 3 from line 1.  Thus,

 

   5x - y + z = 6
   3x + 2y - 3z = -19
-(5x - 15y + 10z = 25)

----------------------------

         14y - 9z  =  -19 (We need this for the next round of elimination)

 

Round 2

 

11y - 9z = -34

14y - 9z = -19

 

Step 5

 

Subtract line2 from line 1.  Thus, 

 

   11y - 9z = -34
-(14y - 9z = -19)

------------------

   -3y = -15

 

y = 5

 

Step 6

 

Substitute y into any of the equations in Step 5,  Thus, 

 

11(5) - 9z = -34

55 - 9z = -34

-9z = -89

 

z = 89/9

 

Substitute y and z into any of the original first 3 equations,  Thus,

 

3x + 2y - 3z = -19

 

3x + 2(5) -3(89/9) = -19

 

3x + 10 - 89/3 = -19

 

9x + 30 - 89 = -57

 

9x - 59 = -57

 

9x = 2

 

x = 2/9

 

Answer:  {x, y, z} = {2/9, 5, 89/9}

 

Hope this helps!

 

Answer 2

Solving the system using Cramer's Rule:

The determinant D of the equation system is

 

         5  -1  1                      6   -1  1               5   6    1                 5   -1  1                                 6

D =    3   2  -3   = -27   D_x = -19  2  -3       Dy = 3  -19  -3      D_z =    3  2  -3     :det coef is      -19

         1  -3  2                      5  -3  2                1   5   2                  1  -3  2                                  5

 

Same way we build x determinant D_x = -6,   D_y = -135   D_z =-267

 

x = D_x/D = -6/-27 = 0.22    y = D_y/D =-135/-27 = 5   z = D_z/D = -267/-27 = 9.88