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.

--
Cramer's Rule for Three Equations — Moosa Academy

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:

a_1 x + b_1 y + c_1 z = d_1
a_2 x + b_2 y + c_2 z = d_2
a_3 x + b_3 y + c_3 z = d_3

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
D 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.
x = \frac{D_x}{D} \qquad y = \frac{D_y}{D} \qquad z = \frac{D_z}{D}

As before, D \neq 0 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:

9x + 4y + 4z = 53
2x + 8y - 3z = -18
-3x + 3y + 6z = 3
Step 1 — the coefficient determinant:
D = 9(57) - 4(3) + 4(30) = 513 - 12 + 120 = 621
Since D \neq 0 , the rule applies.
Step 2 — replace the x -column with the constants:
D_x = 53(57) - 4(-99) + 4(-78) = 3021 + 396 - 312 = 3105
Step 3 — replace the y -column:
D_y = 9(-99) - 53(3) + 4(-48) = -891 - 159 - 192 = -1242
Step 4 — replace the z -column:
D_z = 9(78) - 4(-48) + 53(30) = 702 + 192 + 1590 = 2484
Step 5 — divide each one by D :
x = \dfrac{3105}{621} = 5 , y = \dfrac{-1242}{621} = -2 , z = \dfrac{2484}{621} = 4
⟹ x = 5, y = −2, z = 4

Check in the first equation: 9(5) + 4(-2) + 4(4) = 45 - 8 + 16 = 53 ✓

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 z without finding x 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.
231. Arithmetic Sequences and Series
← Back to Semester