Linear Systems: Consistent or Not

Linear Systems: Consistent or Not

What is it?

A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after elimination). Unique solution: square, full rank, det != 0. Infinitely many: free variables (rank < number of unknowns, still consistent). Homogeneous Ax=0 always has the trivial x=0; extra solutions when det=0. Meaning: the shape of the row-reduced picture is the story.

In simple words

• A system Ax = b.
• Consistent: at least one solution.
• Inconsistent: none (a row 0 = 1 after elimination).
• Homogeneous is never inconsistent. Non-homogeneous can be.

Important terms

LINEAR SYSTEMS: CONSISTENT OR NOT: A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after elimination). Unique solution: square, full rank, det != 0. Infinitely many: free variables (rank < number of unknowns, still consistent). Homogeneous Ax=0 always has the trivial x=0; extra solutions when det=0. Meaning: the shape of the row-reduced picture is the story.

1. THREE CASES: Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.

2. HOMOGENEOUS: Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).

What you should learn

• State the main ideas and notation used in linear systems: consistent or not.
• Apply the rules of linear systems: consistent or not in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• Unique
• Free variable
• Contradiction

THREE CASES

Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.

HOMOGENEOUS

Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).

Full school-level explanation

A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after elimination). Unique solution: square, full rank, det != 0. Infinitely many: free variables (rank < number of unknowns, still consistent). Homogeneous Ax=0 always has the trivial x=0; extra solutions when det=0. Meaning: the shape of the row-reduced picture is the story.

What the last rows say:

WHAT — LINEAR SYSTEMS: CONSISTENT OR NOT

A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after elimination). Unique solution: square, full rank, det != 0. Infinitely many: free variables (rank < number of unknowns, still consistent). Homogeneous Ax=0 always has the trivial x=0; extra solutions when det=0. Meaning: the shape of the row-reduced picture is the story.

RULE — 1. THREE CASES

Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.

METHOD — 2. HOMOGENEOUS

Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).

WORKED EXAMPLE

Worked tell: after elimination you see 0 0 1 | 4. that is z = 4, fine if z is an unknown but 0 0 0 | 4 means 0=4, inconsistent read the bar: coefficients | right-hand side

THREE CASES

Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw the three cartoons in your head before you compute.

Remember

• pivots vs unknowns

Examples

• Unique
• Free variable
• Contradiction

HOMOGENEOUS

Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).

Remember

• always consistent

Examples

• Trivial solution
• Free variables if singular
• Null space named

How to answer a linear systems: consistent or not question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• consistent = at least one x
• 0 = 1 means inconsistent
• 0=4 is death
• Ax=0 always has 0
• pivots vs unknowns
• always consistent
• consistent = at least one x
• 0 = 1 means inconsistent

How the ideas connect

Worked tell: after elimination you see 0 0 1 | 4. that is z = 4, fine if z is an unknown but 0 0 0 | 4 means 0=4, inconsistent read the bar: coefficients | right-hand side

Deeper understanding

Go beyond memorising Linear Systems: Consistent or Not. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked tell: after elimination you see 0 0 1 | 4. that is z = 4, fine if z is an unknown but 0 0 0 | 4 means 0=4, inconsistent read the bar: coefficients | right-hand side

Common mistakes

Exam focus

Core Idea

1. THREE CASES

2. HOMOGENEOUS

Worked Example

Exam focus

Quick revision

• consistent = at least one x
• 0 = 1 means inconsistent
• 0=4 is death
• Ax=0 always has 0
• LINEAR SYSTEMS: CONSISTENT OR NOT: A system Ax = b. Consistent: at least one solution. Inconsistent: none (a row 0 = 1 after eliminatio
• 1. THREE CASES: Unique: n pivots for n unknowns. Infinite: fewer pivots, no bad row. None: a row [0 0 0 | 1]. Draw t

Homogeneous is never inconsistent. Non-homogeneous can be.

For long answers on Linear Systems: Consistent or Not: define clearly → explain the main parts → give one example → state one exam caution.