Solutions to three variable system 2  Algebra II  Khan Academy  Summary and Q&A
TL;DR
The video explains how to determine whether a system of three equations with three unknowns has no solutions or infinite solutions.
Questions & Answers
Q: How can we determine whether a system of three equations with three unknowns has no solutions or infinite solutions?
By eliminating variables one by one, we can pair equations together and check if they intersect or if the solution is nonsensical. If the equations are parallel planes, the system has no solutions.
Q: What is the purpose of eliminating variables in a system of equations?
By eliminating variables, we can simplify the system to have fewer unknowns. This makes it easier to determine the number of solutions.
Q: Can we solve for the unknown variables in a system with no unique solution?
No, because a system with no solution means that the equations represent parallel planes in three dimensions that do not intersect. There is no unique point that satisfies all three equations.
Q: How does multiplying an equation by a negative or positive number help in elimination?
Multiplying an equation by a negative or positive number allows us to manipulate the coefficients of the variables to cancel them out when adding or subtracting equations. This simplifies the system and helps determine the solutions.
Summary & Key Takeaways

The video teaches a method to solve systems of three equations with three unknowns by eliminating variables one by one.

Two equations can be paired to eliminate one variable, leaving two equations with two unknowns.

If the remaining equations do not intersect or have a nonsensical solution, the system has no solutions.