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
A solution of a system is a point that makes every equation true at once. For two variables, write the solution as an ordered pair. One equation gives a line, so a system asks which points the lines share.
Two lines can meet in three ways. If they cross at one point, the system has exactly one solution. If they are parallel and distinct, they share no point, so there is no solution. If both equations draw the same line, every point on it works, so there are infinitely many solutions. A system with at least one solution is consistent; a system with none is inconsistent. Equations that draw the same line, like x + y = 3 and its triple 3x + 3y = 9, are dependent.
Solve x + y = 7 and x - y = 3. Adding gives 2x = 10, so x = 5. Then 5 + y = 7, so y = 2. Alternatively, subtract the second equation from the first: (x + y) - (x - y) = 7 - 3 gives 2y = 4, so y = 2 and x = 5. The solution is (5, 2).
Elimination adds or subtracts equations, sometimes after scaling one of them. For example, to solve x + 2y = 7 and x - y = 1, double the second to 2x - 2y = 2 and add: 3x = 9, so x = 3 and y = 2. In a two-variable system, 0 = 5 is a contradiction with no solution, while 0 = 0 is an identity that can describe the same line.
A free variable can take any value, written as a parameter. In x - y = 0 and x + y + z = 6, let z = t. Then x = y, so 2x + t = 6 and x = y = (6 - t) / 2: one solution for each t. Two equations can fix at most two unknowns, so a consistent system in three unknowns leaves at least one free. A contradiction means no solution.
Solve 3x + y = 14 and x - y = 2. Adding gives 4x = 16, so x = 4. Then x - y = 2 gives y = 2, and both equations check.
One crossing is one solution, parallel lines give none, and a shared line gives infinitely many.
Original lesson and assessment text by the Open Source Taxonomy project.
The rewritten readings were written for this dataset. Another reading is adapted from a Khan Academy article, and the videos link to Khan Academy. Practice questions are by the Open Source Taxonomy project.
2 · Watch
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Where it sits
14 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.