Row operations and geometric interpretation of solution sets | Year 12

01 / LEARN

The idea, step by step

Preserve equivalence

You may swap rows, multiply a row by a nonzero scalar, or add a multiple of one row to another. Each operation applies to every entry, including the augmented constant. These operations preserve the solution set.

A pivot is a leading nonzero coefficient in row-echelon form. Solve pivot variables in terms of free variables. A reduced identity coefficient matrix gives a unique solution immediately.

Interpret the result

A row $[0 0 0 ∣ c]$ with nonzero c means no solution. A zero row adds no restriction. For a consistent system in three unknowns, three pivots give a point, two give a line, and one gives a plane.

Record enough operations to justify the result, even if technology helps with arithmetic. A calculator output should be checked in the original system, not merely substituted into a transformed equation containing a possible transcription error.

A common trap

Change the coefficients but leave the augmented constant unchanged.

Instead: The constant is part of the equation and must undergo the same row operation.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Apply $R_{2} \leftarrow R_{2} - 2 R_{1}$ to [1,1,1|3] and [2,3,1|7].

Subtract twice every entry in row 1.
The new equation is y-z=1.

The second row becomes [0,1,-1|1].

Example 2

Worked example 2

Interpret rows [1,0,2|3], [0,1,-1|4], [0,0,0|0].

Let the non-pivot variable z=t.
Then x=3-2t and y=4+t.

A line:

(3 - 2 t, 4 + t, t)

Check your understanding

What does [0,0,0|-2] mean?

Show answer

No solution.

It asserts 0=-2.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Apply $R_{2} \leftarrow R_{2} - 3 R_{1}$ to [1,2,-1|4] and [3,7,-2|13].
2 marks
Interpret rows [1,0,0|2], [0,1,0|-1], [0,0,1|3].
2 marks
Interpret rows [1,2,0|5], [0,0,1|1], [0,0,0|0].
2 marks
Interpret a consistent three-variable system with one pivot.
2 marks
Is multiplying a row by zero an equivalence operation?
2 marks
Check (2,1,0) in x+y+z=3, 2x-y+z=3, x-y-z=1.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Apply $R_{2} \leftarrow R_{2} - 3 R_{1}$ to [1,2,-1|4] and [3,7,-2|13].
2 marks
Subtract [3,6,-3|12] entry by entry.
The new equation is y+z=1.
[0,1,1|1]
Interpret rows [1,0,0|2], [0,1,0|-1], [0,0,1|3].
2 marks
Every variable has its own pivot.
Read the augmented constants.
Unique solution (2,-1,3).
Interpret rows [1,2,0|5], [0,0,1|1], [0,0,0|0].
2 marks
Choose y=t.
Then x=5-2t and z=1.
Line $(5 - 2 t, t, 1)$ .
Interpret a consistent three-variable system with one pivot.
2 marks
One independent equation restricts three variables.
Two independent directions remain free.
A plane with two free parameters.
Is multiplying a row by zero an equivalence operation?
2 marks
It destroys the row's constraint.
It cannot be reversed.
No.
Check (2,1,0) in x+y+z=3, 2x-y+z=3, x-y-z=1.
2 marks
The three left sides are 3, 3 and 1.
Each agrees with its original constant.
It is a correct solution.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics vector algebra, vector/Cartesian equations and linear systems. Original explanations and tasks.

Official curriculum source ↗

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