Solving systems of linear equations

The essentials

A system of linear equations such as a₁x + b₁y = c₁, a₂x + b₂y = c₂ can be written as AX = B, where A is the coefficient matrix, X the column of unknowns and B the column of constants. A system is consistent if it has at least one solution and inconsistent if it has none.

If |A| ≠ 0, A⁻¹ exists and the system has the unique solution X = A⁻¹B. If |A| = 0, compute (adj A)B. If (adj A)B ≠ O, the system is inconsistent. If (adj A)B = O, the system may have infinitely many solutions or none, and must be checked further.

Key points

• Matrix method: find |A|, then adj A, then A⁻¹ = (1/|A|) adj A, then X = A⁻¹B.
• For two equations in two unknowns, |A| = 0 means the lines are parallel (no solution) or coincident (infinitely many).
• A homogeneous system AX = O always has the solution X = O; it has other solutions only when |A| = 0.
• Substituting the answer back into every equation is a quick check.

Exam tips

• Questions often ask for a value of k that makes a system have no unique solution: set |A| = 0 and solve for k.
• In three-variable problems, an equation with a missing variable makes the cofactor work shorter.

Common mistakes

• Writing X = BA⁻¹ instead of X = A⁻¹B.
• Concluding "infinitely many solutions" as soon as |A| = 0 without checking (adj A)B.