2.2-05 Back substitution

Turn an echelon system into explicit solutions by reasoning upward from its last useful equation. This practice includes negative and fractional values, several unknowns and systems with a free variable. It emphasizes dependency order and substitution accuracy, then checks how the final result relates to the original equations.

Practice focuses on these skills: choose the starting equation from the system’s dependency structure; recover earlier unknowns after solving later pivot equations; handle negative pivots and exact fractional solutions; express pivot variables in terms of an unrestricted free parameter; interpret identity rows without inventing additional restrictions; verify a candidate against every original equation. These distinctions help connect the notation to the mathematical constraints, so that a calculation can be explained as well as completed.

This back substitution questionset mixes single-select and multi-select tasks. The local progression begins with recognition and interpretation, moves into calculations, and ends with comparisons or error diagnosis. Worked explanations show the reasoning behind each result and address likely misconceptions. Read every alternative carefully in multi-select questions; more than one mathematical statement may hold.

Before starting, review forward elimination and rearranging linear equations. Work through calculations on paper, then compare your reasoning with the explanation. Revisit any item where the method was unclear, even if the final choice was correct. This microlearning prepares you to write complete solutions and distinguish fixed values from free choices within the Complete Linear Algebra Course.