Cramer's Rule for Three Equations

Extending Cramer rule to three unknowns: four 3x3 determinants instead of three 2x2 ones, replacing one column at a time with the constants, and dividing each result by D - with a fully worked system.

--

With three unknowns the method is unchanged — only the count grows. Two equations needed three determinants; three equations need four, each of them 3×3.

Concept The system

Three linear equations in three unknowns give a 3×3 coefficient matrix:

Cramer's rule requires the number of equations to equal the number of unknowns, so a 3×3 system is exactly the right shape.

Theorem Four determinants
Dx Dy Dz the shaded column holds the constants
is the determinant of the coefficients as they stand. For each of the other three, replace one column with the constants and leave the rest untouched.

As before, is required. Every one of the four determinants is evaluated by expanding along the first row.

Example A full three-variable solution

Solve the system:

Step 1 — the coefficient determinant:
Since , the rule applies.
Step 2 — replace the -column with the constants:
Step 3 — replace the -column:
Step 4 — replace the -column:
Step 5 — divide each one by :
, ,
⟹ x = 5, y = −2, z = 4

Check in the first equation: ✓

Note Why the method scales well
The steps are fixed, so the method is easy to program.
Each variable is found independently — you can compute without finding first.
The same pattern extends to 4×4 systems and beyond.
Because the operations repeat, mistakes are easier to spot.
Summary
  1. A three-variable system needs four 3×3 determinants: D, Dx, Dy and Dz.
  2. D uses the coefficients unchanged; check that it is not zero before going on.
  3. For each variable, replace only that variable's column with the constants.
  4. x = Dx/D, y = Dy/D and z = Dz/D.
  5. In the worked example D = 621, giving x = 5, y = −2 and z = 4.
283. Relations and Functions
← Back to Semester