Solve the following system using elimination (aka linear combination) attach an extra sheet, if needed.

Question

{ 3x+2y+4z=11
{ 2x-y+3z=4
{ 5x-3y+5z=-1

Answer 1

3 x + 2 y + 4 z = 11

2 x - y + 3 z = 4

5 x - 3 y + 5 z = - 1

3 x + 2 y + 4 z = 11

Multiply both sides by 5

15 x + 10 y + 20 z = 55

5 x - 3 y + 5 z = - 1

Multiply both equation by 3

15 x - 9 y + 15 z = - 3

15 x + 10 y + 20 z = 55
-
15 x - 9 y + 15 z = - 3
____________________

15 x - 15 x + 10 y - ( - 9 ) y + 20 z - 15 z = 55 - ( - 3 )

10 y + 9 y + 5 z = 58

19 y + 5 z = 58

3 x + 2 y + 4 z = 11
+
2 x - y + 3 z = 4
____________________
5 x + y + 7 z = 15

5 x + y + 7 z = 15
-
5 x - 3 y + 5 z = - 1
______________________

5 x - 5 x + y - ( - 3 y ) + 7 z - 5 z = 15 - ( - 1 )

y + 3 y + 2 z = 15 + 1

4 y + 2 z = 16

Divide both sides by 2

2 y + z = 8

Subtract 2 y to both sides

2 y + z - 2 y = 8 - 2 y

z = - 2 y + 8

Replace this value in equation:

19 y + 5 z = 58

19 y + 5 ∙ ( - 2 y + 8 ) = 58

19 y + 5 ∙ ( - 2 y ) + 5 ∙ 8 = 58

19 y - 10 y + 40 = 58

9 y + 40 = 58

Subtract 40 to both sides

9 y + 40 - 40 = 58 - 40

9 y = 18

Divide both sides by 9

y = 18 / 9

y = 2

z = - 2 y + 8

z = - 2 ∙ 2 + 8

z = - 4 + 8

z = 4

Replace tis values in equation:

3 x + 2 y + 4 z = 11

3 x + 2 ∙ 2 + 4 ∙ 4 = 11

3 x + 4 + 16 = 11

3 x + 20 = 11

Subtract 20 to both sides

3 x + 20 - 20 = 11 - 20

3 x = - 9

Divide both sides by 3

x = - 9 / 3

x = - 3

The solutions are:

x = - 3 , y = 2 , z = 4

Answer 2

My typo:

5 x - 3 y + 5 z = - 1

NOT

Multiply both equation by 3

5 x - 3 y + 5 z = - 1

Multiply both sides by 3

15 x - 9 y + 15 z = - 3