Cramer's Method

Solve a 2x2 linear system with determinants only: build the coefficient matrix, compute D, swap each variable's column for the constants to get Dx and Dy, then divide. Worked on 5x - 6y = 15, 3x + 4y = -29.

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Substitution and elimination both make you rearrange the equations. Cramer's method does not. You write down three determinants, divide, and the answers fall out. Every step is fixed, which is why it turns into an algorithm a computer can run.

Concept The coefficient matrix and its determinant

Read the coefficients straight off the system, keeping every negative sign. For and :

A determinant is the first diagonal minus the second diagonal. Compute first: it is the denominator for every variable, and if it comes out the method cannot be used.

Concept Replace a column to get its determinant
5−6 34 15−6 −294 515 3−29 D Dₓ D y the dashed column is replaced by the constants 15 and −29
The denominator is always . For , remove the -column and drop the constants in its place. For , remove the -column instead. Nothing else changes.
Example The full run

Solve and .

⟹ x = −3, y = −5

Watch the sign in : subtracting removes a positive product, which is what pulls the value down to .

Note Why it is called systematic
Nothing is rearranged at any point — the equations are only ever read, never manipulated.
The same three operations run in the same order for every system.
That fixed pattern is exactly what a computer needs, which is why linear solvers use it.
The one thing to check is ; if there is no unique solution.
Summary
  1. Build the coefficient matrix from the system, keeping the negative signs.
  2. Compute first; it is the denominator for every variable and must not be .
  3. For replace the -column with the constants; for replace the -column.
  4. and .
  5. For , : , , , so , .
283. Relations and Functions
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