BetterGrades Precalculus · Unit 8 · Lesson
Systems in three variables
Solve and classify three-variable linear systems and interpret their solutions as intersections of planes.
Start with the situation
A linear equation in three variables describes a plane, and a system seeks their common point, line, plane, or empty intersection.
Systems combine several conditions into one decision. Graphs, equations, inequalities, and matrices are different views of the same requirement: the final result must satisfy every condition at once.
Prerequisite check
- Solve equations and systems.
- Interpret graphs as solution sets.
- Use organized arithmetic and units.
Explanation
Eliminate one variable from two equation pairs, solve the reduced system, and back-substitute.
A contradiction means no solution; a free variable produces an infinite family.
A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through three-plane gallery, elimination ladder, or another equivalent representation.
What the idea is really doing
A system asks for simultaneous truth. Graphs show common intersections, elimination preserves the solution set, and matrices record the same operations compactly; the representation changes, but the solution condition does not.
This lesson narrows that lens to one goal: solve and classify three-variable linear systems and interpret their solutions as intersections of planes. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.
Plan before calculating
Problem
Solve
- Plan
- Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Eliminate one variable from two equation pairs, solve the reduced system, and back-substitute.
- Conclusion
- Why the check works
- Pairwise subtraction isolates variables.
See the idea in three forms
foundation example
Solve
Solution
Pairwise subtraction isolates variables.
representation example
Meaning .
SolutionNo solution.
This example expresses systems in three variables in a second form.
transfer example
Meaning free variable.
SolutionInfinitely many solutions.
A contradiction means no solution; a free variable produces an infinite family.
Read this graph as text
Systems in three variables · Three-plane gallery. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Pairwise subtraction isolates variables. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Solve and classify three-variable linear systems and interpret their solutions as intersections of planes.
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Pairwise subtraction isolates variables.
Read this graph as text
Systems in three variables · Elimination ladder. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for systems in three variables. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Solve and classify three-variable linear systems and interpret their solutions as intersections of planes.
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for systems in three variables.
Read this graph as text
Systems in three variables · Parameterized solution line. Compare the valid path with the tempting shortcut. The figure shows why performing elimination on only one pair and stopping with insufficient information leads to a false conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Solve and classify three-variable linear systems and interpret their solutions as intersections of planes.
Compare the valid path with the tempting shortcut. The figure shows why performing elimination on only one pair and stopping with insufficient information leads to a false conclusion.
Find the first invalid move
A frequent error is performing elimination on only one pair and stopping with insufficient information.
One equation in three variables.
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10 concrete questions
01One equation in three variables.
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02Meaning .
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03Meaning free variable.
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04Solve
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05Explain why this conclusion is valid: . Use the foundation problem as evidence: Solve .
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06Solve the representation example, then name the feature of systems in three variables that it illustrates: Meaning
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07Correct this reasoning and identify the first unsafe assumption: A frequent error is performing elimination on only one pair and stopping with insufficient information.
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08Connect two representations for this example: Solve . Describe what a graph, table, mapping, or algebraic form would have to show.
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09Create a nearby example by changing one number or condition in this prompt: Meaning free variable. Predict the effect, solve your new example, and compare it with the original.
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10Write a short verification checklist for systems in three variables, then apply it to one worked example from this lesson.
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Connect forward
The next lesson, Matrix notation and operations, uses this result as part of a larger structure.
Source record
Original BetterGrades manuscript, rights-separated references.
- Yoshiwara, Modeling, Functions, and Graphs
- Utah College Algebra
- Stitz and Zeager, Precalculus
No long source passage is reproduced.