Solving three equations in three unknowns

What a 3×3 system is

• A system of equations is a set of equations that must all be true at the same time.
• A 3×3 system has three equations in three unknowns — usually $x$, $y$ and $z$.
• A solution is an ordered triple $(x, y, z)$ that satisfies every equation in the system.
• Solving the system means finding that triple — or showing there is no triple, or infinitely many.

The elimination strategy

• Goal: turn three equations in three unknowns into two equations in two unknowns, then into one equation in one unknown.
• Eliminate the same variable from two different pairs of equations:
  • Add or subtract multiples of two equations so one variable cancels.
  • Do this for a second pair, cancelling the same variable.
• You now have two equations in the two remaining unknowns — a 2×2 system.
• Solve that 2×2 system.
• Back-substitute the two values into any original equation to find the third unknown.

Choosing what to eliminate

• Pick the variable that is easiest to cancel — look for matching or opposite coefficients.
• If no coefficients match, multiply an equation through so they do.
• Label your equations (1), (2), (3) and each new one (4), (5), and say what you did: "(1) − (2) eliminates $z$".

Checking the answer

• Substitute your triple back into all three original equations.
• It must make every equation true — checking only one is not enough.