how can i solve these equations?

Question

x + 3y – z = 2
x – 2y + 3z = 7
x + 2y – 5z = –21

Answer 1

Elimination method :

Given equations are

x + 3y - z = 2        ---> (1)

x - 2y + 3z = 7      ---> (2)

x + 2y - 5z = - 21  ---> (3)

To eliminate the y value subtract (3) from (2).

x - 2y + 3z = 7

x + 2y - 5z = - 21

-    -     +      +

_____________

- 4y + 8z = 28 ---> (4)

To eliminate the y value subtract (2) from (1).

x + 3y - z = 2

x - 2y + 3z = 7

-  +     -     -

___________

5y - 4z = - 5     ---> (5)

Multiply 2 to (5)

5y - 4z = - 5     ---> (5) * 2

then the euqation is,

10y - 8z = - 10  --> (6)

To eliminate the z value add the equations (6)&(4).

- 4y + 8z = 28

10y - 8z = - 10

____________

6y = 18

Divide to each side by 6.

⇒ y = 3.

Substitute the y value in (4).

- 4(3) + 8z = 28

- 12 + 8z = 28

Add 12 to each side.

8z = 28 + 12

8z = 40

Divide to each side by 8.

⇒ z = 40/8 = 5.

Substitute y,z in (1).

x + 3(3) - 5 = 2

x + 9 - 5 = 2

x + 4 = 2

Subtract 4 from each side.

x = 2 - 4

⇒ x = - 2.

Solution x = - 2, y = 3,and z = 5.