Systems of Linear Equations

A system of linear equations is simply a handful of equations that all have to be true at the same time, and solving it means finding the values of the unknowns that make every one of them work. They're called linear because the unknowns show up plainly (no squares, no square roots, no unknowns multiplied together), so with two unknowns each equation graphs as a straight line. They're good for any problem where several conditions have to hold at once, like a total count and a total cost. The general form above looks intimidating, but it's only three kinds of pieces: $x_1, \dots, x_n$ (the unknowns you're solving for), $a_{11}, a_{12}, \dots$ (the plain numbers in front of them, where the little subscripts just say which equation and which unknown) and $b_1, \dots, b_m$ (what each equation has to add up to). It's the same thing you did in algebra class with two equations and two unknowns, written so it works for any number of them; bear with the subscripts.

Geometry in two variables. Each equation in two unknowns is a line, and a solution of the system is a point that sits on every line at once. Think about what two lines can do. They can cross once (one solution), run parallel and never meet (none), or be the same line written two ways (infinitely many). That's why you never get exactly two solutions: if two straight lines share two points, they're the same line, so they share all of them. In three variables each equation is a plane, and the same three outcomes hold. There's an animation further down where you tilt one line and see all three cases happen.

Augmented matrix. Writing $x_1$ and $x_2$ over and over gets tedious, and the variables never actually do anything. All the information is in the numbers. So strip away the variables and keep the numbers, lined up in columns, with a bar where the equals signs were:

Elementary row operations don't change the solutions:

1. Replace a row by itself plus a multiple of another row.
2. Swap two rows.
3. Multiply a row by a nonzero constant.

Think about why. These are the same moves you'd make with the equations themselves. If both equations are true at some point, then adding a multiple of one to the other gives an equation that's still true there, and you can undo the step by subtracting the same multiple back off. Swapping two equations just changes the order you wrote them in. Multiplying an equation by a nonzero number is undone by dividing. Since every step can be reversed, no solutions get created or lost along the way.

A common mistake. It's tempting to think multiplying a row by $0$ is fine, since $0$ is just another constant. It isn't: it wipes out an equation, and there's no way to divide back and get it again. Throwing away an equation can add solutions that weren't there before, which is why rule 3 says nonzero.

A system is consistent if it has at least one solution, and inconsistent if it has none. In the next lesson you'll see how to tell which one you've got just by looking at the matrix.

Balancing chemical equations, network flows (traffic, electricity, money), and fitting curves through data points all come down to solving linear systems.