SOLVE SYSTEM OF 3 EQUATIONS

Problem 1 :

Solve:

x - y + z = 5

3x + 2y - z = -2

2x + y + 3z = 10

Solution:

x - y + z = 5 --- (1)

3x + 2y - z = -2 --- (2)

2x + y + 3z = 10 --- (3)

Let us add (1) and (2),

x - y + z = 5

3x + 2y - z = -2

_______________

        4x + y = 3 --- (4)

Multiply the (2) equation by 3 and add by (3).

(2) × 3 ==> 9x + 6y - 3z = -6

      (3) ==> 2x + y + 3z = 10

                ___________________

                  11x + 7y = 4 ---> (5)

Multiply the (4) equation by 7 and subtract by (5).

(4) × 7 ==> 28x + 7y = 21

   (5) ==> 11x + 7y = 4

                     _______________

17x = 17

x = 1

By applying x = 1 in (5), we get

11(1) + 7y = 4

11 + 7y = 4

7y = -7

y = -1

By applying x = 1 and y = -1 in (1), we get

1 + 1 + z = 5

2 + z = 5

z = 3

So, the solution is (x, y, z) = (1, -1, 3).

Problem 2 :

2x + y - z = 3

x - 3y + z = 7

3x + 5y - 3z = 10

Solution:

2x + y - z = 3 ---> (1)

x - 3y + z = 7 ---> (2)

3x + 5y - 3z = 10 ---> (3)

Let us add (1) and (2),

(1) ==> 2x + y - z = 3

(2) ==> x - 3y + z = 7

             _____________

3x - 2y = 10 ---> (4)

Multiply the (2) equation by 3 and add by (3).

(2) × 3 ==> 3x - 9y + 3z = 21

(3) ==> 3x + 5y - 3z = 10

             _______________

                6x - 4y = 31 ---> (5)

(4) × 2 ==> 6x - 4y = 20

(5) ==> 6x - 4y = 31

             _________________

0 = -11

So, there is no solution.

Problem 3 :

Solve the system.

x + y + z = 2

2x + y - z = -1

x + 2z = 5

Solution:

x + y + z = 2 ---> (1)

2x + y - z = -1 ---> (2)

x + 2z = 5 ---> (3)

Let us subtract (1) and (3),

x + y + z = 2

x + 2z = 5

___________

        y - z = -3 ---> (4)

Subtract (1) and (2),

x + y + z = 2

2x + y - z = -1

______________

     -x + 2z = 3 ---> (5)

Add (3) and (5),

4z = 8

z = 2

By applying z = 2 in (3), we get

x + 2(2) = 5

x + 4 = 5

x = 1

By applying x = 1 and z = 2 in (1), we get

1 + y + 2 = 2

3 + y = 2

y = -1

So, the solution is (x, y, z) = (1, -1, 2).

Problem 4 :

Solve each system.

6x - y + 2z = 8

2x + 3y - z = -9

4x + 2y + 5z = -1

Solution:

6x - y + 2z = 8 ---> (1)

2x + 3y - z = -9 ---> (2)

4x + 2y + 5z = -1 ---> (3)

Multiply the (2) equation by 2 and add by (1).

(1) ==> 6x - y + 2z = 8

(2) × 2 ==> 4x + 6y - 2z = -18

              ___________________

                  10x + 5y = -10 ---> (4)

Multiply the (2) equation by 5 and add by (3).

(2) × 5 ==> 10x + 15y - 5z = -45

(3) ==> 4x + 2y + 5z = -1

                ___________________

                 14x + 17y = -46 ---> (5)

(4) × 14 ==> 140x + 70y = -140

(5) × 10 ==> 140x + 170y = -460

                       _________________

-100y = 320

y = -320/100

y = -16/5

By applying y = -16/5 in (4), we get

10x + 5(-16/5) = -10

10x - 16 = -10

10x = 6

x = 3/5

By applying x = 3/5 and y = -16/5 in (1), we get

6(3/5) + 16/5 + 2z = 8

18/5 + 16/5 + 2z = 8

34/5 + 2z = 8

2z = 8 - 34/5

2z = 6/5

z = 3/5

So, the solution is (x, y, z) = (3/5, -16/5, 3/5).

Problem 5 :

5x - 6y + 2z = 21

2x + 3y - 3z = -9

-3x + 9y - 4z = -24

Solution:

5x - 6y + 2z = 21 ---> (1)

2x + 3y - 3z = -9 ---> (2)

-3x + 9y - 4z = -24 ---> (3)

Multiply the (2) equation by 2 and add by (1).

(1) ==> 5x - 6y + 2z = 21

(2) × 2 ==> 4x + 6y - 6z = -18

                     ____________________

                    9x - 4z = 3 ---> (4)

Multiply the (2) equation by 3 and subtract by (3).

(2) × 3 ==> 6x + 9y - 9z = -27

(3) ==> -3x + 9y - 4z = -24

         ___________________

9x - 5z = -3 ---> (5)

Subtract (4) and (5),

z = 6

By applying z = 6 in equation (4).

9x - 4(6) = 3

9x - 24 = 3

9x = 27

x = 3

By applying x = 3 and z = 6 in (1), we get

5(3) - 6y + 2(6) = 21

15 - 6y + 12 = 21

27 - 6y = 21

-6y = -6

y = 1

So, the solution is (x, y, z) = (3, 1, 6).