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Grade 12 · lesson 4 of 4

Update the Entire Augmented Row

Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: In an augmented system, R1=[1,1|2], R2=[2,3|3]. Replace R2 with R2−2R1. Candidate answers: (A) R2=[0,1|-1] (B) R2=[0,1|1] (C) R2=[0,5|7] (D) R2=[1,2|1].
  • Independent check inputs: In an augmented system, R1=[1,1|3], R2=[2,3|4]. Replace R2 with R2−2R1. Candidate answers: (A) R2=[0,1|-2] (B) R2=[0,1|1] (C) R2=[0,5|10] (D) R2=[1,2|1].

Learn the idea

Matrices organize linear operations and systems through rows and columns. Multiplication uses row-column products, a determinant tests two-dimensional invertibility, and valid row operations preserve a system’s solution set when applied to every entry. Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row. The right-side entry participates just like every coefficient.

Here is the method on the supplied teaching case. Subtract twice every entry of R1, including its augmented value: [2−2,3−2|3−2(2)]=[0,1|-1]. Such a row operation preserves the system’s solution set.

Leaving the augmented value unchanged changes the system. Apply the operation to every entry, keep the source row available until the calculation is complete, and interpret resulting rows as equivalent equations.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: In an augmented system, R1=[1,1|2], R2=[2,3|3]. Replace R2 with R2−2R1. Candidate answers: (A) R2=[0,1|-1] (B) R2=[0,1|1] (C) R2=[0,5|7] (D) R2=[1,2|1]. Solution: Subtract twice every entry of R1, including its augmented value: [2−2,3−2|3−2(2)]=[0,1|-1]. Such a row operation preserves the system’s solution set. The final answer is R2=[0,1|-1]. Leaving the augmented value unchanged changes the system. Apply the operation to every entry, keep the source row available until the calculation is complete, and interpret resulting rows as equivalent equations.

Your turn

Check your reasoning

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Helpful tips
  • Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.
  • Leaving the augmented value unchanged changes the system.
In an augmented system, R1=[1,1|3], R2=[2,3|4]. Replace R2 with R2−2R1.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

R2=[0,1|-2]

Subtract twice every entry of R1, including its augmented value: [2−2,3−2|4−2(3)]=[0,1|-2]. Such a row operation preserves the system’s solution set.

Take it further

Changed-input task: In an augmented system, R1=[1,1|9], R2=[2,3|10]. Replace R2 with R2−2R1. Candidate answers: (A) R2=[0,1|-8] (B) R2=[0,1|1] (C) R2=[0,5|28] (D) R2=[1,2|1]. Show the mathematical reason, not only the selected answer. Expected result: R2=[0,1|-8]. Subtract twice every entry of R1, including its augmented value: [2−2,3−2|10−2(9)]=[0,1|-8]. Such a row operation preserves the system’s solution set.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. In an augmented system, R1=[1,1|4], R2=[2,3|5]. Replace R2 with R2−2R1.
Need a hint?
  • Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.
  • Leaving the augmented value unchanged changes the system.
2. In an augmented system, R1=[1,1|5], R2=[2,3|6]. Replace R2 with R2−2R1.
Need a hint?
  • Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.
  • Leaving the augmented value unchanged changes the system.
3. In an augmented system, R1=[1,1|6], R2=[2,3|7]. Replace R2 with R2−2R1.
Need a hint?
  • Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.
  • Leaving the augmented value unchanged changes the system.

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