a system equations

Question

3x-5y-2z=-5

6x+2y+3z=5

4x-3y-z=8

x=? ;y=? ; z=?

Answer 1

Elimination method :

Given equations are 3 x - 5 y - 2 z = - 5 ---> (1)

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

4x - 3 y -  z = 8 -----> (3)

Multiply each side by - 6  in eq ( 1)

We get the equation - 18 x + 30 y +12 z  = 30  ---------> ( 4 )

Multiply each side by 3  in eq (2)

We get the equation 18 x + 6y + 9 z  =15 ---------> ( 5 )

To eliminate the x  value add  (4 )&(5).

- 18 x + 30 y +12 z  = 30

18 x + 6y + 9 z  =15

________________

36 y + 21 z  =  45

12 y + 7 z  =  15 ------> (6)

Multiply each side by - 4  in eq (2 )

We get the equation - 24 x - 8 y -12 z  = - 20 --------> (7)

Multiply each side by 6 in eq (3)

We get the equation 24 x - 18 y - 6 z  = 48 --------> ( 8 )

To eliminate the x  value add (7)&(8).

- 24 x - 8 y -12 z  = - 20

24 x - 18 y - 6 z  = 48

_________________

- 26 y -18 z = 28

- 13 y - 9 z = 14 -----> (9)

Multiply each side by  7 in eq (6 ) 

We get the equation 108 y + 63  z = 135  -------> ( 10)

Multiply each side by 7  in eq (9) 

We get the equation 91 y- 63 z = 98 -------> (11 )

To eliminate the z value add (10)&(11).

108 y + 63  z = 135

- 91 y - 63 z = 98

_____________

17 y = 233

y = 233 / 17

Substitute the y  value in eq ( 6 ).

12 y + 7 z =15

12( 233 / 17 ) + 7 z = 1 5

2796 / 17 + 7 z = 1 5

7z = 15 - 2796 /17

7z  = (255-2796) / 17

7z = - 2541 / 17

z =(- 2541) / (17 * 7)

z = - 363 / 17

Substitute the values z  & y values in eq ( 1 )

3x -5 y-2 z = - 5

3x - 5 (233 / 17)  -  2(-363/17) = - 5

3x -(1165 / 17) + (726/17) = - 5

3 x - (1165+ 726)/17  =- 5

3x - 439 /17 = - 5

3x  = - 5+ 439 /17 

x  = 354/17 *3

x  =118/17.

Solution of the system of  the equations is x = 118 / 17 , y  = 233 / 17 41 and  z = - 363 / 17.