Solving Systems with Matrices

Explore augmented matrix, elementary row operations, back-substitution.

Lesson 3 of 53 parts Practice Flashcards

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Augmented Matrix

Objective: Explain augmented matrix and use it to solve or interpret a problem.

An augmented matrix places system coefficients next to constants in consistent variable order; it stores equations compactly. A swapped column would represent a different system.

Worked examples

Try itSolving Systems with Matrices — practice 1.1: Find top-left entry of [[3,7],[1,2]]+[[5,1],[3,4]].Show answer

Step 1: Add corresponding top-left entries 3+5. Step 2: The resulting top-left entry is 8. Step 3: Matrix dimensions match: both are 2×2.

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Augmented Matrix

Objective: Explain augmented matrix and use it to solve or interpret a problem.

An augmented matrix places system coefficients next to constants in consistent variable order; it stores equations compactly. A swapped column would represent a different system.

Worked examples

Try itSolving Systems with Matrices — practice 1.1: Find top-left entry of [[3,7],[1,2]]+[[5,1],[3,4]].Show answer

Step 1: Add corresponding top-left entries 3+5. Step 2: The resulting top-left entry is 8. Step 3: Matrix dimensions match: both are 2×2.

Elementary Row Operations

Objective: Explain elementary row operations and use it to solve or interpret a problem.

Swap rows, scale a row by a nonzero factor, or add a multiple of one row to another to preserve the system solution set. It destroys information and cannot be reversed.

Worked examples

Try itSolving Systems with Matrices — practice 2.1: Row [1,3] is multiplied by 4. What is its second entry?Show answer

Step 1: Multiply every entry in the row by 4. Step 2: The second entry is 4·3. Step 3: It becomes 12.

Back-Substitution

Objective: Explain back-substitution and use it to solve or interpret a problem.

After obtaining a triangular or reduced form, solve known entries and substitute to find remaining variables, then check the original system. Entry or arithmetic errors can occur during row operations.

Worked examples

Try itSolving Systems with Matrices — practice 3.1: Solve x+y=15, x−y=7. Find x.Show answer

Step 1: Add equations: 2x=22. Step 2: Divide by 2: x=11. Step 3: Back-substitution gives y=4.

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