Systems of three variables

Math

Definition

A set of three linear equations with three unknowns (x, y, z). The solution is an ordered triple (x, y, z) that satisfies all three equations simultaneously. These can be solved using elimination, substitution, or matrices.

How It Works

  1. Choose a variable to eliminate first.
  2. Use two pairs of equations to create two new equations with only two variables.
  3. Solve the resulting 2×2 system.
  4. Substitute back to find the eliminated variable.

Examples

  • x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 → solution is (1, 2, 3)
  • Finding three unknown prices when given three different combination totals

Related Terms

Elimination method Substitution method Solving systems with matrices Matrix operations

Study This Concept

Practice systems of three variables with free review games in these units:

Algebra 2 Unit 2: Linear Functions and Systems