How do I solve the system of linear equations?

Question

3x-3y+6z=15

x-2y-z=8

5x-8y+13z=22

If system is dependent express x, y and z in terms of parameter a

Answer 1

The system of equations is

3x - 3y + 6z = 15 (1)

x - 2y - z = 8 (2)

5x - 8y + 13z = 22 (3)

We can use elementary methods (elimination and substitution) to solve the problem. We can also use a matrix method such as Gauss Elimination and Back Substitution.

First, we shall use elimination and Substitution.

Add (1) and -3*(2).

3x - 3y + 6z -3(x - 2y - z) = 15 - 3*8

3x - 3y + 6z - 3x + 6y + 3z = -9

3y + 9z = -9

y + 3z = -3 (4)

Add (3) and -5*(2).

5x - 8y + 13z - 5(x - 2y - z) = 22 - 5*8

5x - 8y + 13z - 5x + 10y + 5z = -18

2y + 18z == -18

y + 9z = - 9 (5)

Subtract (5) from (4).

y + 3z - (y + 9z) = -3 -(-9)

-6z = 6

z = -1

From (5), obtain

y - 9 = -9

y = 0

From (2), obtain

x - 2(0) - (-1) = 8

x + 1 = 8

x = 7

Answer: x=7, y = 0, z = -1

If you need a matrix solution, let me know.