Solve a system of three linear equations in three variables by systematically eliminating one variable at a time.
Three unknowns, three equations, one strategy: reduce to something you already know how to solve. Pick a variable to eliminate, and combine two DIFFERENT pairs of equations to knock it out both times -- this leaves you with two equations in only two variables, exactly last week's skill. Solve that pair, then substitute back twice to recover the third variable.
The bookkeeping matters more than any new idea. Label your equations (1), (2), and (3) so you always know which pair you combined and why. A clean plan: eliminate the same variable from (1)&(2) and from (1)&(3), solve the resulting 2-variable system, then plug both answers into any ORIGINAL equation to find the last unknown.
Mastery looks like: They reduce a 3-variable system to a 2-variable system by systematic elimination, solve it, and verify all three values in every original equation.
Common stumbles: Combining the same pair of equations twice (which cannot eliminate a second variable); forgetting to check the final answer in all three equations.