Solve a 3-variable system using algebraic methods—elimination, substitution, or the matrix method (Cramer’s Rule / inverse). Graphically, each equation is a plane in 3D; a unique solution is the common point of intersection.
General Form
a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3x + b3y + c3z = d3
Methods at a Glance
| Method | When to Use | Core Steps | Mini Example (outline) |
| Elimination | Coefficients align or can be aligned | Eliminate one variable by combining two equations → reduce to 2 equations in 2 variables → solve → back-substitute for the third variable. | From E1 & E2 eliminate z; from E1 & E3 eliminate z → solve the 2×2 result → back-substitute. |
| Substitution | One equation isolates a variable easily | Solve one equation for x (or y, z) → substitute into other two → solve 2×2 system → back-substitute twice. | From E1: x = (d1 − b1y − c1z)/a1 → put into E2, E3 → solve → back-substitute. |
| Matrix (Cramer / Inverse) | Clear numeric coefficients; check det ≠ 0 | Write AX = B. If det(A) ≠ 0, then X = A−1B or use Cramer’s Rule: x = Δx/Δ, y = Δy/Δ, z = Δz/Δ. | Compute Δ = det(A); replace columns to get Δx, Δy, Δz → divide. |
Worked Mini-Example
E1: x + y + z = 6 E2: 2x − y + z = 3 E3: x + 2y − z = 3
Eliminate z using E2 − E1 ⇒ x − 2y = −3; use E3 − E1 ⇒ y − 2z = −3.
Solve with E1 to get: x = 9/7, y = 15/7, z = 18/7.
Nature of Solutions (Consistency)
- Unique solution: three planes meet at one point (Δ ≠ 0).
- No solution: planes are parallel or mutually inconsistent.
- Infinitely many solutions: planes coincide or intersect along a line.
Prefer elimination for hand calculations; use the matrix method for clean coefficient matrices and to test uniqueness via det(A).
