Chapter 3
Systems of Equations and Inequalities
Two numbers add to twelve. That fact alone leaves plenty of pairs, and no amount of staring picks out the one that is meant. Add a second fact, that their difference is four, and the field of possibilities collapses. Conditions work like clues: each one narrows what is still allowed, and what matters in the end is what survives all of them at once. When the conditions pile up faster than you can juggle them, how would you find out what is left?
What You'll Explore
5 lessons.
- Systems in Two Variables
Picture two straight lines on the same grid. They look like they have only two options, meeting somewhere or never meeting, but counting the possibilities carefully is more interesting than that. This lesson settles how many answers a pair of linear conditions can have, and how the algebra tells you which case you are in.
- Systems in Three Variables
Adding a third unknown means adding a third equation, and the picture lifts off the flat page entirely. What does a single equation even look like once there are three variables, and what can three such conditions share? This lesson carries elimination into that larger setting and asks what the possible answers become.
- Systems of Inequalities
Real limits are rarely exact: a budget says at most, a schedule says at least. A condition like that does not pick out one point; it leaves a whole region of possibilities open. This lesson looks at what survives when several such limits apply at once, and at what you might want to find inside what is left.
- Matrices and Systems of Equations
Write out an elimination in full and the bookkeeping starts to feel repetitive, since the variable names come along for the ride while the numbers do all the work. Could you drop the letters entirely and push a grid of numbers around instead? This lesson tests that idea and asks which moves on such a grid are legal.
- Determinants and Cramer's Rule
Solve a general system once, with letters standing in for every coefficient, and the answer arrives as a formula rather than a pair of numbers. One quantity built from those coefficients keeps appearing inside it. This lesson asks what that number is measuring, and what happens in the case where it turns out to be zero.