Solve system: 2x+3y+2z=1, x+4y-z=7, 3x+y+3z=-2

[mjh94]

Solve the system

2x+3y+2z=1
x+4y-z=7
3x+y+3z=-2
In need help My teacher WILL NOT

 

[pka]

Re: Can anyone help?!?!?!?!

I would use matrices.
What methods do you have to use?

 

[nycfunction]

Solve the system

2x+3y+2z=1
x+4y-z=7
3x+y+3z=-2
In need help My teacher WILL NOT

We have a system of linear equations in 3 variables.
It's easier to use matrix algebra but I will use the elimination method this time.

I will give each equation a number.

2x+3y+2z=1...equation 1
x+4y-z=7...equation 2
3x+y+3z=-2...equation 3

I will multiply equation 2 by -2 and then add it to equation 1. This will delete the x variable from equation 1.

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

-2x - 4y + 2z = -14...We add this equation to equation 1.

-2x - 4y + 2z = -14
2x + 3y + 2z = 1

-y + 4z = -13...I will call this one equation A.

I will now multiply equation 2 by -3 and add it to equation 3 to delete the x variable from equation 3. Got it?

-3(x + 4y -z) = -3(7) becomes -3x -12y + 3z = -21

I now add -3x -12y + 3z = -21 to equation 3 to delete the x variable from equation 3.

3x+y+3z=-2 plus -3x -12y + 3z = -21

-3x -12y + 3z = -21
3x +y +3z = - 2

-11y + 6z = -23...This I will call equation B

Notice that equations A and B have become a system of linear equations in 2 variables.
See it?

Here they are:

-y + 4z = -13...equation A
-11y + 6z = -23...equation B

I can now multiply equation A by -11 and then add it to equation B to do away with the y variable.

-11(-y + 4z) = (-11)(-13)

11y -44z = 143...add this equation to equation B
-11y +6z = -23

-38z = 120

z = 120/-38

z = (-60/19)

We found z. We now need to find x and y.

I will plug the value of z I just found into EITHER equation A or B to find y.

I selected equation A and it is

-y + 4z = -13

-y + 4(-60/19) = - 13

-y -240/19 = -13

-y = -13 + 240/19

-y = -7/19

y = 7/19.... We now know the values of y and z.

The last step is to select any of the original equations (1, 2 or 3) and plug the values of y and z to find x and we are done.

I chose equation 1 and it is

2x + 3y+ 2z = 1...We need to find x. So plug and chug.

2x + 3(7/19) + 2(-60/19) = 1

2x + 21/19 - 6_6/19 = 1

2x - 5_4/19 = 1

2x = 1 + 5_4/19

2x = 6_4/19

x = 6_4/19 divided by 2

x = 3_2/19

Solution: (x, y, z) = (3_2/19, 7/19, - 60/19)