Solve the following system of equations in three variables using algebraic methods only. Check your results.

Question

6x + 2y = 22

 

3y - z = 5

 

2x - 3z = 3

Answer 1

Solve:

6x + 2y = 22
3y - z = 5
2x - 3z = 3

 

Rewrite:

   6x   +   2y          = 22           [eq1]

               3y  -  z   =   5           [eq2]

   2x              - 3z   =  3           [eq3]

 

Now, to use the Elimination Method:

   6x    +   2y            =   22       [eq1 again]

               27y    -9z   =   45       [9*eq2]

  -6x                 +9z  =   -9       [(-3)*eq3]

 ---------------------------------------------------   {elimination; add equations]

              29y             =   58         

                 y              =    2          [eq4]              [divide both sides by 29]

 

               -3y             =   -6         [(-3)*eq4]

                3y     - z     =    5         [eq2 again]

    --------------------------------     [elimination; add equations]

                         -z     =   -1

                          z     =     1       [eq5]

 

                       3z   =      3        [4*eq5]

   2x              - 3z   =      3        [eq3 again]

------------------------------------     [elimination; add equations]

   2x                       =      6

    x                        =      3

 

Check:

     Does    6*(3) + 2*(2) = 22   ?

                    18 + 4 = 22   ?

                        22 = 22   ?yes

     Does     2*(3) - 3*(1) = 3   ?

                    6  -  3  = 3   ?

                       3 = 3   ?yes

     Does     2*(3) - 3*(1) = 3   ?

                    6  -  3   =  3   ?

                       3 = 3   ?yes

Answer 2

 6x + 2y = 22

 3y - z = 5

 2x - 3z = 3




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


 Multiplies the third equation by -3. The system becomes:

 6x + 2y = 22

 3y - z = 5

-6x + 9z = -9
---------------------------------------------

Adds that to equation #1. The system is now:

2y + 9z = 13

6x + 2y = 22

 3y - z = 5


Now pairs that up equations #1 and #3

2y + 9z = 13

3y - z  = 5


Multplies equation by #1 by 3 and equation #3 by -2

6y + 27z = 39

-6y + 2z = -10

Then adds them together
29z = 29

z=1

Subsitutes into Equation #3:
   3y - z = 5

   3y - 1 = 5
  
   3y = 6

    y = 2

Substitutes into original equation #1

6x + 2y = 22

6x + 2(2) = 22

6x + 4 = 22

6x = 18

x = 3

The check is left for you.

Answer 3

6x + 2y = 22    Divide both sides by 2.

3x + y = 11 (equation 1)

3y - z = 5

3y - 5 = z  (equation 2)

2x - 3z = 3   Replace z with equation 2.

2x - 3(3y-5) = 3

2x - 9y + 15 = 3

2x - 9y = -12  (equation 3)

Multiply both sides of equation 1 by 9.

9(3x + y = 11)

27x + 9y = 99  (equation 4)

Add equation 3 and 4 together.

  2x - 9y = -12

27x + 9y = 99

        29x = 87

            x = 3    In equation 1, replace x with 3.

3(3) + y = 11

    9 + y = 11

          y = 2    Replace y with 2 in equation 2.

   3y - 5 = z

3(2) - 5 = z

    6 - 5 = z

         1 = z         (3,2,1)