Solving Systems with Matrices

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

Lesson 2 of 63 parts Practice Flashcards

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Augmented Matrix

Objective: Explain augmented matrix, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$a x+b y=c ↔ [a b | c]$

Worked examples

Try itThe equation 3x+5y=7 is written as an augmented-matrix row. What is its last entry?Show answer

Step 1: An augmented row lists variable coefficients followed by the right-hand constant. Step 2: The row is [3, 5 | 7]. Step 3: The last entry is 7, not a variable coefficient.

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Augmented Matrix

Objective: Explain augmented matrix, apply the rule, and justify relevant restrictions.

Formulas

$a x+b y=c ↔ [a b | c]$

Worked examples

Try itThe equation 3x+5y=7 is written as an augmented-matrix row. What is its last entry?Show answer

Step 1: An augmented row lists variable coefficients followed by the right-hand constant. Step 2: The row is [3, 5 | 7]. Step 3: The last entry is 7, not a variable coefficient.

Elementary Row Operations

Objective: Explain elementary row operations, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$Rᵢ←kRᵢ, k≠0; Rᵢ←Rᵢ+kRⱼ$

Worked examples

Try itRow [1,5] is multiplied by 3. What is its second entry?Show answer

Step 1: Multiply every entry in the row by 3. Step 2: The second entry is 3·5. Step 3: It becomes 15.

Back-Substitution

Objective: Explain back-substitution, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

A common solution satisfies every equation simultaneously

Worked examples

Try itSolve x+y=12, x−y=2. Find x.Show answer

Step 1: Add equations: 2x=14. Step 2: Divide by 2: x=7. Step 3: Back-substitution gives y=5.

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