Systems of equations
Two conditions hold at once, each on the same unknowns. Which values satisfy both, and how do you find them exactly?
Where you are 7 inside
What this area is about
One linear equation in x and y, such as x + y = 10, is true for a whole line of pairs. A system adds a second condition, and its solution is every pair that makes both true at once. For two linear equations in two unknowns that is one pair, none (parallel lines) or infinitely many (the same line). This area finds those pairs by graph and by algebra, then widens to three unknowns, curves and inequalities.
How these fit together
The idea. Solving systems by graphing defines a solution as a crossing point and shows the three cases.
Procedures. Substitution puts one equation’s expression for a letter into the other. Elimination adds or subtracts equations, scaled if needed, so one letter cancels.
Extensions. Systems in three variables eliminate one letter to get a two-variable system. Systems with a curve meet a line with a parabola, which can happen twice, once or not at all. Systems of linear inequalities answer with a shaded region instead of a point.
Using it. Systems in word problems turn two totals, such as a count and a value, into two equations.
A path through it
Graphing first. Then substitution, and systems of inequalities. After substitution: elimination and systems with a curve. After elimination: word problems and three-variable systems.
After this: matrices, which write a whole system as one equation, and fitting a quadratic to data, which solves a three-variable system.