Matrices
Cramer's rule 3 × 3 worksheets
Solve twelve three-variable systems with Cramer's rule. Use exact determinants and rank evidence for unique, infinitely many and no-solution systems.
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Cramer's rule 2 × 2 worksheets
Solve twelve two-variable systems with Cramer's rule. Practice unique, infinitely many and no-solution systems with exact determinant reasoning.
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Write the coefficient determinant D and the replacement determinants Dx and Dy. If D is nonzero, calculate x = Dx/D and y = Dy/D. If D is zero, compare the coefficient and augmented systems to decide between infinitely many solutions and no solution.
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Customize this worksheetLet the main determinant choose the path
Build D from the coefficient matrix, then replace one coefficient column at a time with the constants to form Dx and Dy. When D ≠ 0, each variable is its replacement determinant divided by D, kept as an exact integer or fraction.
When D = 0, Cramer's quotients are undefined and the system needs a consistency check. Row-reduce the augmented matrix: equal ranks below the number of variables mean infinitely many solutions; a larger augmented rank means no solution.
Learning objective
What this activity builds
Compare the coefficient determinant with column-replacement determinants, then solve or classify each two-variable system.
Worked example
Solve x + y = 5 and 2x − y = 1.
x = 2 and y = 3.
D = −3, Dx = −6 and Dy = −9, so x = Dx/D = 2 and y = Dy/D = 3.
Teaching notes
Make the practice count
- Keep each equation in the same variable order before building matrices.
- Replace only one coefficient column at a time with the constants.
- When D is nonzero, divide each replacement determinant by D and keep the result exact.
- When D is zero, check consistency instead of dividing by zero.
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Choose one or more solution types. The preview and PDF use the same systems and the separate key shows determinant work plus rank evidence for singular cases.
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