Cramer's Rule

Algebra 2 · Axiom Academy

Solve a linear system with pure determinants: — elegant in theory, with real limits in practice. 1. The Setup: Systems and Determinants Consider a system of two linear equations in two unknowns. Its coefficients form a matrix A , and the constants form a column : The key object is the determinant of A — a single number built from the two diagonals. When it is not zero, the system has exactly one solution. Where does the formula come from? Solve the general 2×2 system by elimination and the determinant pattern appears on its own. Step 1 — multiply the first equation by d and the second by b : Step 2 — subtract to cancel the identical bdy terms: The denominator ad-bc is exactly , and the numerator ed-bf is the determinant of A with its first column replaced by the constants . Solving for y the same way replaces the second column instead: where D_x and D_y come from D by replacing the x - or y -column with the constants. This needs . Watch the constants column slide in to replace a variable's column — the determinant of that new matrix, divided by , gives the variable: Worked example — solve the system Check: 2(1)+3(2)=8 and 4(1)-2=2 . ✓︎ 4. Extension to Larger Systems The column-replacement idea works for any size. For a 3×3 system: The same rule holds — divide each column-replaced determinant by the main determinant — but now the determinant itself takes real work. A 3×3 expands into three 2×2 determinants:

This is the written version of the interactive lesson above. See the full Algebra 2 course.