Solving a 3 by 3 System of Equations (the most organized way)  Summary and Q&A
TL;DR
Learn how to solve a 3+3 system of equations using elimination, choosing the lowest common multiple of the coefficients.
Questions & Answers
Q: How do you determine which variable to eliminate first in a 3+3 system of equations?
The variable to eliminate first in a 3+3 system is determined by choosing the lowest common multiple of the coefficients of that variable. The goal is to make the coefficients of the chosen variable the same, with alternating signs.
Q: Can you explain the process of elimination in a 3+3 system of equations?
In a 3+3 system, elimination involves multiplying the equations by the LCM of the coefficients of the chosen variable. This makes the coefficients of the chosen variable the same, with alternating signs. The resulting system can then be combined to eliminate the chosen variable and solve for the remaining variables.
Q: What should be done if the coefficients of the variables don't share a common multiple in a 3+3 system?
If the coefficients of the variables don't share a common multiple, it is not possible to eliminate a variable. In this case, another method, such as substitution or matrix methods, should be used to solve the system.
Q: Is it necessary to choose the variable with the lowest coefficients for elimination in a 3+3 system?
No, it is not necessary to choose the variable with the lowest coefficients for elimination in a 3+3 system. The choice of variable for elimination depends on finding the lowest common multiple of the coefficients, which may not always correspond to the variable with the lowest coefficients.
Summary & Key Takeaways

To solve a 3+3 system of equations, choose the lowest common multiple (LCM) of the coefficients of one variable as the elimination factor.

Multiply the equations by the LCM to make the coefficients of the chosen variable the same, with alternating signs.

Combine the equations to eliminate the chosen variable and solve the resulting 2variable system using elimination again.

Substitute the found values into one of the original equations to solve for the remaining variable.

The final solution for the 3+3 system is obtained by combining the values of all three variables.