find the unknowns usuing either elimination or substitution 2x+6y+3z=9 , 5x-3y-5z=3 , 4x+3y+2z=3

Question

Solve for all three unknowns

Answer 1

Elimination method

The equations are 2x + 6y + 3z = 9 -----> (1)

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

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

To eliminate the y value add the equations (2) & (3).

5x - 3y - 5z = 3

4x + 3y + 2z = 3

____________________

9x - 3z = 6 -----> (4)

Multiple to each side of equation (2) by 10x - 6y - 10z = 6 -----> (5)

To eliminate the y value add the equations (1) & (5).

2x + 6y + 3z = 9

10x - 6y - 10z = 6

____________________

12x - 7z = 15 ----> (5)

To get two equations 4 and 5 that contain opposite terms multiply the fourth equation by 7 and fifth equation negative 3.

Write the equations in column form and add the corresponding columns to eliminate z - variable

63x - 21z = 42

-36x + 21z = -45

__________________

27x = -3

x = -1/9

Substitute the x value in equation (4).

9(-1/9) - 3z = 6

- 1 - 3z = 6

3z = -7

z = -7/3

Substitute the x,z value in equation (1).

2(-1/9) + 6y + 3(-7/3) = 9

-2/9 + 6y - 7 = 9

6y = 16 + 2/9

6y = (144 + 2)/9

y = 146/54

y = 73/27

Solution x = -1/9 , y = 73/27 and z = -7/3.