Undergrad Solving System of Equations w/ Gauss-Jordan Elimination

[Oannes]

I am fairly new here so I apologize for any mistakes in my post.

My question concerning solving a system of equations using Gauss-Jordan Elimination is specifically about different ways to handle a possible constant. Say for instance you have three equations:

1. X₁+X₂+X₃ + 3 = 9
   
2. 2X₁+4X₂+X₃ = 13
3. X₁+2X₂+2X₃ = 11

Now of course you could just subtract 3 from the answer and go from there and that is obviously the simplest and proper way to do it, but I am curious as to if there are other ways. For example, could you perhaps take out the constant of 1, then perform the elimination, then do some sort of operation to get the correct answer?. If all of the equations had the same constant you would be able to take it out and then add/subtract it back into your answer to get the correct value of each variable. This got me to thinking if it would be possible to take all the constants present, divide it by the number of variables, then add/subtract it back into your solved variables to get the correct answer? I think not since the constants are independent to each equation and are not related. Still, I was curious if anyone else has thought about this and if there is any way to solve something like this besides just taking the constant from the sum before performing elimination.

Note: I am sorry if this post is jumbled, I am just trying to hop into this community and I decided to post on something that I have been thinking on even though it seems kind of trivial. Thanks to everyone in advance for any sort of discussion, I hope to become active in this community as I find many of these topics very interesting.

 

[BvU]

could you perhaps take out the constant of 1

what constant would that be, specifically ?

take all the constants present, divide it by the number of variables

could you work out in detail what you mean ? 'it' ? or 'them' ? or 'all of them' ?

 

[Mark44]

Now of course you could just subtract 3 from the answer and go from there and that is obviously the simplest and proper way to do it, but I am curious as to if there are other ways. For example, could you perhaps take out the constant of 1, then perform the elimination, then do some sort of operation to get the correct answer?. If all of the equations had the same constant you would be able to take it out and then add/subtract it back into your answer to get the correct value of each variable. This got me to thinking if it would be possible to take all the constants present, divide it by the number of variables, then add/subtract it back into your solved variables to get the correct answer? I think not since the constants are independent to each equation and are not related. Still, I was curious if anyone else has thought about this and if there is any way to solve something like this besides just taking the constant from the sum before performing elimination.

This doesn't make much sense.
"Subtract 3 from the answer..." -- the "answer" will be a triple of numbers.
"Take all the constants present, divide it by the number of variables..." Who is "it" here? Does "it" refer to the sum of all the constants? In any case, adding the constant terms and then dividing this sum by the number of equations will give you a number that is unrelated to anything.

There is a standard technique, Gauss-Jordan elimination, for solving systems of equations. You must have at least heard of this, as you have it in your thread title. This technique entails writing the equations in what is called an augmented matrix, and then reducing the matrix to give solutions for the variables.

The underlying geometry for the system you showed is three planes in three-dimensional space. If there is a unique solution, this means that all three planes intersect at only a single point. Other possibilities are that the planes don't intersect at all (possibly all of the planes are parallel to each other) or that the planes intersect in a line, in which case there are an infinite number of solutions; namely, all of the points on the line of intersection.

 

[FactChecker]

I can't understand what you are saying, but here is how to proceed:
In all cases, do the same thing to both sides of an equality. If you subtract the left side of equation 3 from the left side of equation 1, then subtract the right side of equation 3 from the right side of equation 1 (left side always equals right side, so you are doing the same thing to both sides, just in a different form.). What you do on the left side for the Gauss-Jordan elimination, do the same thing on the right side using the right side constants.

If you still have trouble, you should post your work and ask about the specific step that you are having trouble with.