Row operations and geometric interpretation of solution sets | Year 12

01 / LEARN

Understand the idea.

Preserve equivalence

Elementary row operations preserve a system's solution set: interchange rows, multiply a row by a nonzero scalar, or add a multiple of one row to another. Apply the operation to the entire augmented row, including the constant.

Reduce systematically to expose pivots, then solve pivot variables in terms of any free variables. Record enough row operations to justify the result. Even when technology performs arithmetic, a final substitution in the original equations checks that the intended system was entered.

Interpret the result

A row $[0 0 0 ∣ c]$ states $0 = c$ . If $c \neq 0$ , the entire system is inconsistent. If $c = 0$ , the row is redundant and does not restrict any variable. A zero row alone does not specify how many free variables remain.

For a consistent system in three unknowns, three pivots fix a point, two leave a line, and one leaves a plane. Choose a parameter for each free variable and write all coordinates so the geometry of the complete solution set is visible.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

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

Worked solution

1
Apply the operation entry by entry.
$R_{2} - 2 R_{1} = [2 - 2 (1), 3 - 2 (1), 1 - 2 (1) ∣ 7 - 2 (3)]$
The augmented constant undergoes the same subtraction as every coefficient.
2
Simplify the replacement row.
$R_{2} \leftarrow [0, 1, - 1 ∣ 1]$
The first coefficient is eliminated while the remaining entries become $1, - 1, 1$ .
3
Interpret the new equation.
$y - z = 1$
The row records coefficients of $x, y, z$ followed by the right-hand constant; the first row is unchanged.

Answer

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

EXAMPLE 02

Worked example 2

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

Worked solution

1
Translate the nonzero rows into equations.
$x + 2 z = 3, y - z = 4$
Each row gives one linear restriction on the variables.
2
Identify the free variable.
$z = t, t \in ℝ$
There is no pivot in the third column, and the zero row adds no restriction.
3
Solve the pivot variables.
$x = 3 - 2 t, y = 4 + t$
Substitute the free variable into the two nonzero row equations.
4
State the line of solutions.
$(x, y, z) = (3 - 2 t, 4 + t, t) = (3, 4, 0) + t (- 2, 1, 1)$
Two independent conditions in three variables leave one free direction.

Answer

A line: $(3 - 2 t, 4 + t, t)$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

No solution.

The augmented row $[0, 0, 0 ∣ - 2]$ asserts $0 = - 2$ . That contradiction cannot hold for any values of the variables, so the system has no solution.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 $x + y + z = 3$ , $2 x - y + z = 3$ , $x - y - z = 1$ .
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 $x = 5 - 2 t$ 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 $x + y + z = 3$ , $2 x - y + z = 3$ , $x - y - z = 1$ .
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 ↗

In this lesson

Understand the idea.

Preserve equivalence

Interpret the result

See the method in action.

Worked example 1

Apply the operation entry by entry.

Simplify the replacement row.

Interpret the new equation.

Worked example 2

Translate the nonzero rows into equations.

Identify the free variable.

Solve the pivot variables.

State the line of solutions.

Try it yourself.

Check your understanding

Keep practising