From System to Matrix

Algebra 2 · Axiom Academy

Turn a linear system such as into a single augmented matrix — coefficients aligned by variable, constants past the bar. 1. A System Is Two Facts at Once A system of equations is two or more equations that share the same variables. Its solution is the set of values that makes every equation true at the same time — geometrically, the single point where the lines cross. Our worked example — two equations, two unknowns 2. Strip It to an Augmented Matrix To build the augmented matrix , lift the numbers out of the equations and drop them into a grid. Each column collects the coefficients of one variable; the constants go into a final column, set off by a bar. Coefficients of x fill the first column. Coefficients of y fill the second column. A vertical bar stands where the equals signs were. Constants fill the last column. If a variable never appears in an equation, its coefficient is 0 — and you still write that 0 so the columns stay lined up. Here the second equation has no y , so a 0 holds its place in the y -column: 3. A Row Move Is an Equation Move Because each row is an equation, anything you may legally do to the equations you may do to the rows. There are three such moves, and none of them changes the solution. Reorder the equations. Which line you write first is arbitrary; the solution is untouched. Multiply an equation through by a nonzero number — that multiplies every entry in the row by the same number.

This is the written version of the interactive lesson above. See the full Algebra 2 course.