Linear Systems with Two Variables

A system of equations is simply a group of equations that all have to be true at the same time. Here there are two, each a straight line (that's why they're called "linear"), and all we're really asking is which pair of numbers $x$ and $y$ works in both at once. That's how you answer questions with two unknowns, like how many adult and child tickets were sold. The formula up top looks like a lot of subscripts, but each line is just a straight-line equation: $a_1$ and $b_1$ (how much $x$ and $y$ count in the first equation) and $c_1$ (what they have to add up to). The little 1s and 2s only tell you which equation a number belongs to. That's all it is.

So far you've solved one equation in one unknown. Here's the key idea for two: if you can get rid of one variable, you're back to that familiar problem. Both methods below are just two ways of doing that.

Substitution. Solve one equation for one variable, then plug that expression into the other equation. Now the other equation has only one unknown. This is easiest when some variable already has coefficient 1, so solving for it doesn't create fractions.

Elimination. Multiply the equations so one variable has opposite coefficients, then add them. Why is that allowed? If two equations are both true, their sum is true too, so adding them can't lose the solution. Choose the multiples well and one variable cancels out.

Three possible outcomes. Think about what two lines can do.

• One solution: the lines cross once.
• No solution: the lines are parallel and never meet. Elimination ends in a false statement like $0 = 5$, which is the algebra telling you no point works.
• Infinitely many: both equations are the same line in disguise. Elimination ends in $0 = 0$, true but useless, because every point on the line works.

It's always worth checking your answer in both original equations; it takes ten seconds.

Break-even analysis (cost equals revenue), mixing two products to hit a target, and supply meeting demand are two-equation systems.