help solving the system of equations

Question

solve the system of equations

 

 3x + 4y + z = 7

         2y + z = 3

-5x + 3y + 8z = -31

 

I have 2y + z = 3

      =  2y - 2y + z = 3 - 2y

      =  z = 3 - 2y

      =  z = (-2y) + 3

 

 

3x + 4y + (-2y) + 3 = 7

 3x + 2y = 4

 

 

 

Answer 1

 3x + 4y + z = 7..........(1)

         2y + z = 3..........(2)

-5x + 3y + 8z = -31.....(3)

 

Let us first eliminate z from equations (1) and (3) to get equations (4) and (7) with two variables.

 

(1) - (2) gives us

3x + 2y = 4 ......(4)

 

multiply (2) by 8 on both sides

16y + 8z = 24 .....(5)

 

multiply (3) by -1 on both the sides

5x -3y -8z = 31 .....(6)

 

Add equations (5) and (6) to eliminate z

5x + 13y = 55 ......(7)

 

Let us eliminate x from equations (4) and (7) to find the value of y

multiply (4) by -5

-15x -10y = -20 .....(8)

 

multiply (7) by 3

15x + 39y = 165 .....(9)

 

Add (8) and (9) to eliminate x

29y = 145 .....divide both sides by 29

y = 5

 

Substitute y=5 in (4)

3x + 2y = 4 .....(4)

3x + 2.5 = 4

3x + 10 = 4 subtract 10 from both the sides

3x = -6  divide both sides by 3

x = -2

 

Substitute y = 5 in (2)

2y + z = 3 .....(2)

2.5 + z = 3

10 + z = 3 ....subtract 10 from both the sides

z = -7

 

Solution: x = -2, y = 5, and z = -7