Cramer's rule and Gauss elimination
Solve a system with determinants (Cramer), then by row operations (Gauss), and see how a zero determinant gives parallel lines.
- UK: Year 12
- Iran: grade 12
- Afghanistan: grade 11
- Syria: grade 12
- Palestine: grade 11
- Russia: grade 10
Guide
Cramer's rule: for ax + by = e and cx + dy = f let D be the determinant of the coefficients. Dx is D with the x column replaced by the right-hand sides, and Dy the same for y. Then x = DxD and y = DyD. It needs D ≠ 0.
Gauss elimination: write the system as an augmented matrix (coefficients, then the right-hand sides). Use row operations (swap, multiply a row, add a multiple of a row) to get zeros under the leading entries; then back-substitute, or continue to the reduced form.
If D = 0 the two lines have the same slope: they are parallel (no solution) or the same line (infinitely many).
Ask your child why row operations do not change the solution of the system.