solving systems of linear equations in four variables

[RachelB]

I need help with solving systems of linear equations in four variables. The problem is as follows:

2w - x + 5y + z = -3
3w + 2x + 2y - 6z = -32
w + 3x + 3y - z = -47
5w - 2x - 3y + 3z = 49

The direction for this problem say "solve the system of equations. Describe what you are doing at each step in your solution process."

 

[soroban]

Hello, RachelB!

I need help with solving systems of linear equations in four variables

. . \(\displaystyle \begin{array}{cccc}2w\,-\,x\,+\,5y\,+\,z & = & -3 \\
3w\,+\,2x\,+\,2y\,-\,6z & = & -32 \\w\,+\,3x\,+\,3y\,-\,z & = & -47 \\5w\,-\,2x\,-\,3y\,+\,3z & = & 49\end{array}\)

It would help of you told us what methods are available to you:
Elimination, Substitution, Cramer's Rule, Augmented Matrices, etc.

 

[RachelB]

I can use any algebraic method to find the solution.
By any, I mean linear combination method or substition method because this are the only methods that I have learned in school so far.
I think they prefer us to use the linear combination method.
Thank you for helping!!!

 

[Denis]

2w - x + 5y + z = -3
3w + 2x + 2y - 6z = -32
w + 3x + 3y - z = -47
5w - 2x - 3y + 3z = 49

What do you get if you add the 3rd and 4th equations together?

 

[Denis]

You emailed me , saying: what do you mean, I don't understand.
Took me a while to "remember" who you were!!

I sent you this as answer:

You said you understood linear combinations.

w + 3x + 3y - z = -47
5w - 2x - 3y + 3z = 49
These 2 equations are added together this way:
w + 5w = 6w
3x + (-2x) = x
3y + (-3y) = 0
-z + 3z = 2z
-47 + 49 = 2 (forgot to include this one)
So this leaves:
6w + x + 2z = 2

Start by learning how to solve equations with 2 variables.
Can't teach by email: go here:

http://www.mathnstuff.com/math/algebra/asystem.htm