Types of Solution Sets

Linear Algebra (Matrices) · Axiom Academy

A linear system has exactly one of three outcomes — one solution, none, or infinitely many — and the row-reduced matrix tells you which. Treat each equation in two variables as a line in the plane. A solution is a point lying on every line at once. With two lines there are only three ways they can sit relative to each other — and each one is a different kind of solution set. 2. One Solution: Lines That Cross When two lines have different slopes they meet at exactly one point — that point is the unique solution. Watch the two lines settle and the single crossing light up. Why "unique" is guaranteed here The coefficient matrix has determinant . A nonzero determinant means full pivots, which means each variable is pinned down — so the answer is forced and one of a kind. 3. No Solution: Parallel Lines Give two lines the same slope but different intercepts and they run parallel forever — there is no point on both, so the system is inconsistent . The animation slides the second line up off the first; the gap never closes. same left side, two different totals — impossible Any RREF row of the form with kills the system. Here (the rows are not independent), and the rank of the coefficient matrix is smaller than the rank of the augmented matrix — the algebraic fingerprint of "no solution". 4. Infinitely Many: The Same Line Twice

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.