Systems of Linear Equations and Their Solution Sets
A linear system has no solution, exactly one, or infinitely many, and nothing else. Reading which case you are in, and describing an infinite solution set, is the starting point for the whole subject.
What a learner can do afterwards
- Classify a two-variable system by the geometry of its lines
- Describe an infinite solution set with a free parameter
- Give an inconsistent system and say what makes it inconsistent
1 · Read
You already know a solution to two linear equations is where graphs cross, one point. A system gathers equations and asks which points make every one true at once. To test a pair, try it in each equation: 2 and negative 1 make 3 x minus y equals 7 and x minus 2 y equals 4 true. A pair failing even one equation is not a solution.
Two lines leave three options: one crossing is one solution, parallel gives none, and a shared line gives infinitely many. x plus y equals 1 beside x plus y equals 2 cannot both hold. x plus y equals 4 beside 2 x plus 2 y equals 8 share every point.
Solve x plus y equals 7 and x minus y equals 3. Add to kill y, giving 2 x equals 10, so x equals 5. Put 5 in the first equation, giving y equals 2. Check: 5 plus 2 is 7 and 5 minus 2 is 3. For a shared line, name it with a free parameter like y equals 4 minus x.
Elimination adds or subtracts whole equations to kill one variable. A result of 0 equals 5 means parallel lines and no solution, while 0 equals 0 means one shared line. Two equations in three unknowns leave one variable free, so the answer is none or infinitely many, never exactly one.
One crossing is one solution, parallel lines give none, and a shared line gives infinitely many.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.