Solving Systems in Three Variables
A system in three variables needs three equations, and its solution is an ordered triple that satisfies all three. The strategy is to shrink the problem down.
Eliminate one variable from a pair of equations, then from another pair, leaving a two-variable system you already know how to solve. Back-substitute to recover the third variable.
Reduce to two variables
Pick a variable to eliminate. Combine two of the equations to cancel it, then combine a different pair to cancel the same variable again.
Now you have two equations in two variables. Solve that smaller system with elimination or substitution.
Back-substitute for the triple
With two variables known, substitute them into any original equation to find the third. That gives the full ordered triple.
Verify by plugging into all three original equations — a correct solution satisfies every one.
Worked examples
Example 1: the strategy in brief
Outline how to solve a system in .
Answer: Reduce, solve, back-substitute
Example 2: verifying a triple
How do you confirm solves a 3-variable system?
Answer: It satisfies all three equations
Try one yourself
Common questions
How many equations do I need for three variables?
Three independent equations. Fewer leaves the system underdetermined with infinitely many solutions.
What is the overall strategy?
Eliminate one variable to drop to a two-variable system, solve that, then back-substitute to find the eliminated variable.
What does the solution look like?
An ordered triple — the single point (in the one-solution case) satisfying all three equations.
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