# I keep getting fractions trying to solve Systems of Linear Equations with 3 variables

1. Feb 12, 2009

### Raizy

(Edited on Friday 13th 2:53 p.m.)

1. The problem statement, all variables and given/known data

ANYTHING that looks similar to solving these by addition/elimination:

Equation One: 3x - y + 2z = 1
Equation Two: 2x +3y +3z = 4
Equation Three: x + y - 4z = -9

The book's answers: (-1, 0, 2)

The attempt at a solution

Okay here is what I did in order to observe: Re-do the problem 3 times in 3 different ways.

Method 1: Eliminate X first -- in which I got the correct answers.
Method 2: Eliminate Y first -- in which I got the correct answers.
Method 3: Booboo. I got a fraction when the first Y was solved for. I got y = 42/116 or 21/58

Comments: So why does it get messed up when I decide to eliminate Z first? How do I know which variable I should eliminate first in order to get the correct answer?

Step 1. Add equation 1 to equation 2:

1a.--> 3(3x - y + 2z) = 3(1)
1b.--> 9x -3y + 6z = 3

2a.--> -2(2x +3y +3z) = -2(4)
2b.--> -4x -6y -6z = -8

Add equation 1b and 2b to get equation 4: 5x -9y = -8 (Edit at 3:52 p.m. Okay... hmm I just picked this error up...

Step 2: Add equation 1 to equation 3:

1a.--> -4(3x -y +2z) = -4(1)
1c.--> -12x +4y -8z = -4

3a.--> -2(x +y -4z) = -2(-9)
3c.--> -2x -2y +8z = 18

Add equation 1c and 2c to get equation 5: -14x +2y = 14

Step 3: Treat equations 4 and 5 as if it were a system of two systems of linear equations.

4a.--> -14(5x -9y) = -14 (-8)
4b.--> -70x +126y = 112

5a.--> -5(-14x +2y) = -5(14)
5b.--> 70x -10y = -70

Add equation 4b to 5b to get the Y variable: Y=42/116 or 21/58

Does it matter which variable you choose to solve for? What concepts could I be missing? I've re-read the instructions several times -- I've spent 3 days already trying to re-try with no luck. I am sure, unless my eyes are playing tricks on me, is that it is stated it should not matter which variable you eliminate first as you will always get the same answer.

Last edited: Feb 13, 2009
2. Feb 12, 2009

### AssyriaQ

Re: I keep getting fractions trying to solve Systems of Linear Equations with 3 varia

Hi Raizy,

Can you show how you are trying to solve this? If you show us, we can help you pinpoint where the problem is.

3. Feb 12, 2009

### symbolipoint

Re: I keep getting fractions trying to solve Systems of Linear Equations with 3 varia

Try eliminating the 'y' using the last two equations. You would then have a system of two equations and two unknowns.

Do you want to try matrix operations instead?

4. Feb 13, 2009

### HallsofIvy

Re: I keep getting fractions trying to solve Systems of Linear Equations with 3 varia

No one can tell what you are doing wrong if you don't show us what you are doing! The roots of these equations are, in fact, simple whole numbers. It's probably most likely that you are just making arithmetic errors. Perhaps you are not subtracting negative numbers correctly. That's a very common error.

5. Feb 13, 2009

### Raizy

Re: I keep getting fractions trying to solve Systems of Linear Equations with 3 varia

yeah, sorry folks for not showing my work. I was assuming I had some universal error which always resulted me getting fractions after solving for the first X Y or Z for systems of linear equations in 3 variables questions. I am currently in class right now so I'll bump this thread next time I stumble on a problem.

6. Feb 13, 2009

### Raizy

Re: I keep getting fractions trying to solve Systems of Linear Equations with 3 varia

Impossible, I really don't like reading ahead in the book without mastering the previous section. Unless this is one of those cases where my life would get easier?

FINAL EDIT: Okay, I did a simple mistake... I think I got it (hopefully).

Last edited: Feb 13, 2009