C
CraftExam
medium difficulty8 min

Consistency of Linear Equations

A system of linear equations is consistent if it has at least one solution. It is inconsistent if it has no solution.

In one line: Non-zero determinant gives one answer; zero determinant needs a consistency test.

Practice This Concept

Understand it

For AX = B, determinant methods help test solution type. If |A| ≠ 0, the system has a unique solution. If |A| = 0, use adj(A)B to distinguish cases: when adj(A)B is a non-zero matrix, the system is inconsistent; when adj(A)B is a zero matrix, the system may have infinitely many solutions or needs further examination depending on the equations.

Example

For equations x + y = 2 and 2x + 2y = 5, A = [[1,1],[2,2]], so |A| = 0. The left sides are proportional, but constants are not proportional; hence the system has no solution.

Formula and rule

For AX = B: if |A| ≠ 0, unique solution.

If |A| = 0 and adj(A)B is non-zero, no solution.

If |A| = 0 and adj(A)B is zero, the system is consistent with infinitely many solutions or requires further equation-level verification.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Test consistency of x + y = 3 and 2x + 2y = 6.

  1. 1A = [[1,1],[2,2]], B = [[3],[6]]. |A| = 1×2 - 1×2 = 0.
  2. 2The second equation is exactly twice the first, including the constant.
  3. 3Therefore the equations represent the same line and have infinitely many solutions; the system is consistent.

Method you can reuse

  1. 1Write the system as AX = B.
  2. 2Compute |A|.
  3. 3If |A| ≠ 0, state unique solution.
  4. 4If |A| = 0, compare dependence of equations or compute adj(A)B.
  5. 5Then classify as inconsistent, infinitely many solutions, or needing further verification.

Visual tip

No diagram is required. For two-variable systems, line graphs can optionally show intersecting, parallel, or coincident lines.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

If |A| ≠ 0 for AX = B, what type of solution does the system have?

Think first, then show answer
It has a unique solution.

Common wrong answer

Saying the inverse exists when |A| = 0 and trying to solve by X = A−1B.

Don't mix this up

Assuming |A| = 0 always means infinitely many solutions. It may also mean no solution.

Exam tip

When |A| = 0, compare both coefficients and constants or use adj(A)B; do not declare the result from determinant alone.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to check consistency, find values of a parameter for which a system has unique or non-unique solutions, or classify solution type.
  1. 1

    Enough for 1 mark

    A system of linear equations is consistent if it has at least one solution. It is inconsistent if it has no solution.

  2. 2

    Add for 2 marks

    Example: For equations x + y = 2 and 2x + 2y = 5, A = [[1,1],[2,2]], so |A| = 0. The left sides are proportional, but constants are not proportional; hence the system has no solution.

  3. 3

    Add for 3 marks

    For AX = B, determinant methods help test solution type.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept