Choose the most efficient method for a system (graphing, substitution, elimination) and classify systems with no solution or infinitely many.
Three tools now solve any linear system: graphing (best for a quick visual estimate), substitution (best when a variable is already alone, like y = ...), and elimination (best when coefficients already match or are easy to scale into matching). Strong algebra students do not have a favorite - they glance at the equations and pick whichever tool costs the fewest steps.
Sometimes eliminating a variable eliminates BOTH variables at once, leaving only numbers. If that leftover statement is false (like 0 = 12), the system has no solution - the two lines are parallel and never meet. If it is true (like 0 = 0), the system has infinitely many solutions - the two equations describe the very same line. Neither outcome is a mistake; it is the algebra correctly reporting what the lines actually do.
Mastery looks like: They classify a system as one-solution, no-solution, or infinite-solutions from the elimination result alone, and justify it in terms of the lines.
Common stumbles: Panicking when a variable disappears instead of reading what the leftover statement says; assuming "no numbers left" always means no solution.