use pure elimination method to solve

Question

2x + y - z =6

3x -2y +3z =-7

x + y + z =1

Answer 1

To Eliminate a variable, we must find/make equations that have the same (or opposite) coefficients for that variable.

 

    2x   +   y    -    z   =   6            [eq1]

    3x   -   2y  +  3z   =  -7            [eq2]

      x   +   y   +   z   =    1            [eq3]

 

After looking for ways to get the same coefficients, I decided to create a modified eq3 and subtract it from eq2; that eliminates both x and z:

    3x   -   2y    +   3z   =   -7       [eq2]

    3x   +  3y    +   3z   =    3       [3*eq3]

 ---------------------------------      [Elimination; subtract equations]

           -  5y                =  -10

               y                 =      2      [divide by (-5); eq4]

 

Now, I observe that eq1 and eq3 may be added to eliminate z:

    2x   +   y    -    z   =   6            [eq1]
      x   +   y   +   z   =    1            [eq3]

 -------------------------------          [Elimination; add equations]

    3x   +  2y             =   7           [eq5]

               2y             =   4             [2*eq4]

  ------------------------------          [Elimination; subtract equations]

    3x                        =   3 

      x                        =   1           [eq6]

 

Now, we can eliminate x and y (use any starting eq1-eq3):

      x   +   y   +   z   =    1            [eq3]

      x                       =    1            [eq6 again]

 ------------------------------           {Elimination; subtract equations]

                y   +   z   =    0           [eq7]

                y             =    2           [eq4 again]

 -------------------------------         [Elimination; subtract equations]

                          z   =   -2          [eq8]

 

Check:

  Does       2x   +   y    -    z   =   6     ?       [eq1]

                2(1) +  2    -  (-2)   =   6     ?

                            6 = 6   ?yes

  Does      3x   -   2y  +  3z   =  -7     ?       [eq2]

               3(1) - 2(2) +  3(-2)  = -7   ?

                            -7  = -7   ?yes

  Does      x   +   y   +   z   =    1    ?           [eq3]

               (1)  +  (2)  +   (-2)  =  1   ?

                       1   =   1   ?yes

                         

Answer 2

 

2x + y - z =6 ______ eqn 1
3x -2y +3z =-7 ____ eqn 2
x + y + z =1 ______ eqn 3

 

Let's eliminate the y from eqn 1 and eqn 2.

Multiply eqn 1 by 2 and add to eqn 2

 

4x + 2y - 2z = 12

3x - 2y + 3z = -7

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

7x + z = 5

 

Now eliminate y from eqn 2 and 3

Multiply eqn 3 by 2 and add to eqn 2

 

3x -2y +3z =-7

2x + 2y + 2z = 2

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

5x + 5z = -5

 

This can be reduced to:

x + z = -1

 

7x + z = 5

 x + z = -1     subtract

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

6x  = 6

x = 1

 

Plug x = 1 into x + z = -1

1 + z = -1

z = -2

 

In one of the original equations

replace x with 1 and z with -2

and solve for y