How to Solve a System of Equations in 3 Variables (without Matrices)
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The problem: Solve the system of equations:
x+2y-z=-4
-2x+4y-z=6
2x+2y+3z=1
This problem is long, please don't get half way and give up. It takes some time, but it can be done.
The solution: The ordered triple (-3, 1/2, 2) or x=-3; y=1/2 and z=2.
To solve a system of equation in 3 variables you will need to first choose which variable you will want to eliminate first. Then choose two equations and eliminate that variable, giving you a new equation in 2 variables (equation #4). Then choose one of the equations you used to eliminate the first variable and use it with the unused equation to eliminate the same variable giving you a 5th equation (equation #5). Now use equations #4 and #5 to eliminate one of the two variables and solve for the remaining variable. Then use this value to solve for the other variable using either equations #4 or #5. Finally, use one of the original equations to replace the two known values into the equation, and solve for the final variable. After this, you will want to check your answers in the remaining two equations to make sure all three equations give you a true statement.
Make sure to check your work in the end. Some people do not like the extra work, but it is a way that you can ensure your answer is correct. If you realize that your answer is not correct, you will need to check your work and you may even need to rework the problem. This will allow you to compare work against what you have done and find mistakes. Most of my students find that the mistakes they make are minimal, but are highly consequential.