System Solution Types

Algebra 2 · Axiom Academy

One solution, none, or infinitely many — read it from where the lines land and from what row reduction leaves behind. 1. One Solution — Consistent and Independent The lines y = x + 1 and y = -x + 5 have different slopes . Rewrite each in standard form, build the augmented matrix , and watch the geometry and the row reduction resolve together. The lines y = 2x + 1 and y = 2x - 2 share the same slope but different y -intercepts. In standard form their matrices are almost identical — one row operation exposes the contradiction. 3. Infinitely Many — Consistent and Dependent Here x + y = 4 and 2x + 2y = 8 . The second equation is just the first multiplied by 2 , so both describe the same line . What does that leave in the matrix? 4. The Three Cases at a Glance Hold one line fixed and change only the second. Its slope and intercept decide the outcome before you solve a thing — watch the second line move through all three fates. Same slope, different intercept Consistent: at least one solution exists Inconsistent: no solution exists Independent: the equations are not multiples of each other Dependent: one equation is a multiple of the other The row-reduced matrix wears its answer on its sleeve: You can now read a linear system's solution type from its geometry and from its row reduction — no guessing required. Scroll up to revisit any case.

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