When does a system of linear equations have a unique solution?

Exactly when the determinant of the coefficient matrix is not zero. Then Cramer’s rule, the inverse matrix or Gaussian elimination all give the same single solution.

How do I tell "no solution" from "infinitely many solutions"?

Reduce the augmented matrix. A row 0 0 0 | c with c ≠ 0 is a contradiction, so there is no solution. A row of all zeros means one equation is redundant, so there are infinitely many solutions.

What is a non-trivial solution?

A homogeneous system always has the trivial solution x = y = z = 0. A non-trivial solution is any other solution; it exists exactly when the coefficient matrix is singular.

Systems of Linear Equations (DSE M2): Cramer’s Rule and Gaussian Elimination

For a system Ax = b with coefficient determinant Δ: if Δ ≠ 0 there is a unique solution (Cramer’s rule: x = Δₓ/Δ). If Δ = 0 there is either no solution or infinitely many, and you must use Gaussian elimination to find out which.

Solve Systems of Linear Equations questions with AI

The three planes of Example 1 with k = 0 and h = 1. They meet along one line, which is why the system has infinitely many solutions. Try it yourself →

Linear Systems: Rules to Remember

Cramer’s rulex = Δₓ/Δ, y = Δᵧ/Δ, z = Δ_z/Δ (needs Δ ≠ 0)

Δ ≠ 0Unique solution

Δ = 0, some Δₓ, Δᵧ, Δ_z ≠ 0No solution

Δ = Δₓ = Δᵧ = Δ_z = 0No solution or infinitely many: use Gaussian elimination

Inverse-matrix methodx = A⁻¹b (needs |A| ≠ 0)

Homogeneous system Ax = 0Always consistent; non-trivial solutions ⟺ |A| = 0

Unique, No or Infinitely Many Solutions: How to Decide

Find Δ, the determinant of the coefficient matrix, in terms of the parameter.
The system has a unique solution exactly when Δ ≠ 0. Solve Δ = 0 to find the special values of the parameter.
For each special value, substitute it and reduce the augmented matrix by row operations.
If a row becomes 0 0 0 | c with c ≠ 0, there is no solution. If it becomes 0 0 0 | 0, there are infinitely many: let one variable be t and express the others in terms of t.

Geometrically each equation is a plane. A unique solution is one common point; infinitely many solutions means the planes share a line; no solution means they have no common point.

Worked Examples — Systems of Linear Equations

Example 1: A system with a parameter: unique or infinitely many solutions

Consider the system x + y + z = 2, 2x + 3y + z = 3, x + 2y + kz = h. (a) Find the range of k for which it has a unique solution. (b) When k = 0, find h so that the system is consistent, and solve it.

Δ = 1(3k − 2) − 1(2k − 1) + 1(4 − 3) = k
(a) Unique solution ⟺ Δ ≠ 0 ⟺ k ≠ 0
(b) With k = 0: R₂ − 2R₁ gives y − z = −1; R₃ − R₁ gives y − z = h − 2
Consistent only if h − 2 = −1, so h = 1
Let z = t: y = t − 1, x = 2 − y − z = 3 − 2t

Answer: (a) k ≠ 0; (b) h = 1, (x, y, z) = (3 − 2t, t − 1, t) for any real t

Example 2: Cramer’s rule for two unknowns

Solve 2x + y = 5, x − 3y = −1 by Cramer’s rule.

Δ = (2)(−3) − (1)(1) = −7
Δₓ = (5)(−3) − (1)(−1) = −14
Δᵧ = (2)(−1) − (5)(1) = −7
x = −14 / −7 = 2, y = −7 / −7 = 1

Answer: x = 2, y = 1

💡 Exam Traps & Secrets

⚠️ Δ = 0 does not mean "no solution"When Δ = 0 the system has no unique solution. It may still have infinitely many. You must reduce the augmented matrix to tell the two cases apart.

⚠️ Cramer’s rule cannot be used when Δ = 0Dividing by Δ = 0 is undefined. For the special value of the parameter, switch to Gaussian elimination and write the general solution with a parameter t.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Gaussian elimination)Solve by Gaussian elimination: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.

(Parameter and Δ)For what values of k does the system x + y + kz = 1, x + ky + z = 1, kx + y + z = 1 have a unique solution?

(Homogeneous system)Find the non-trivial solutions of x + 2y − z = 0, 2x + y + z = 0, x − y + 2z = 0.