Systems of equations practice
Systems of Three Equations worksheets
Solve three-variable linear systems by true elimination or substitution, including one solution, infinitely many solutions, and no solution.
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Solve each system with the selected method. Give an exact triple, complete parameterization, or explicit contradiction.
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Show the method, not only the classification
Elimination keys retain legal whole-row operations. Substitution keys isolate a variable, use that expression in both remaining equations, and continue the substitution.
Learning objective
What this activity builds
Use exact row operations or recursive substitution to classify and solve systems in x, y, and z.
Worked example
What proves that a system has no solution?
A legal derivation that reaches 0 = c for a nonzero exact number c.
The contradiction shows the original equations cannot all hold at once.
Teaching notes
Make the practice count
- Keep each whole-row operation visible.
- For substitution, put the first isolated expression into both remaining equations.
- Check a unique ordered triple in all three originals.
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Choose your options, update the preview, then download the exact same pages.
Choose elimination or substitution, 4, 6, or 8 systems, and any nonempty selection of solution types. The same exact worksheet object powers preview and PDF.