Systems of three linear equations | Year 12

01 / LEARN

The idea, step by step

Eliminate systematically

Eliminate one variable from two pairs of equations, solve the two-variable system, then back-substitute. Check all original equations; a point satisfying only two may lie on an intersection line but not the third plane.

A coefficient determinant different from zero guarantees a unique solution. A zero determinant only tells you there is no unique solution; consistency still needs checking.

Read dependence geometrically

Every nontrivial linear equation is a plane. Three planes can meet at one point, share a line or plane, or have no common point. Dependent equations repeat information rather than adding another independent restriction.

An equation $0 = c$ with nonzero c is inconsistent. An equation $0 = 0$ is redundant. Introduce a free parameter for each unconstrained variable and describe the whole common solution set.

A common trap

A zero determinant always means no solution.

Instead: It means no unique solution. Row reduction distinguishes inconsistency from free parameters.

02 / WORKED EXAMPLES

See the thinking

Example 1

Worked example 1

Solve x+y+z=6, x-y=0, z=2.

Set z=2 to get x+y=4.
Use x=y to obtain both remaining values.

(2, 2, 2)

Example 2

Worked example 2

Solve x+y+z=1, 2x+2y+2z=2, x-y=0.

The second equation repeats twice the first.
Let x=y=t and solve for z.

(t, t, 1 - 2 t)

for real t.

Check your understanding

Can x+y+z=6 and 2x+2y+2z=13 hold together?

Show answer

No.

Twice the first left side must equal 12, not 13.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Solve x+y+z=9, x-y=1, z=2.
2 marks
Solve x+y=3, y+z=5, x+z=4.
2 marks
Classify x+y+z=2, 2x+2y+2z=5, z=1.
2 marks
Parametrise x+y=2, z=1, 2x+2y=4.
2 marks
Solve x+2y+z=8, y-z=1, x-z=2.
2 marks
What does a unique solution of three independent plane equations represent?
2 marks

Read the worked solutions

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

Solve x+y+z=9, x-y=1, z=2.
2 marks
Write x=y+1.
Then 2y+3=9, so y=3.
$(4, 3, 2)$
Solve x+y=3, y+z=5, x+z=4.
2 marks
Add the first two and subtract the third: 2y=4.
Back-substitute.
$(1, 2, 3)$
Classify x+y+z=2, 2x+2y+2z=5, z=1.
2 marks
Twice the first equation would require left side 4 in the second.
The second constant is 5, giving a contradiction.
No solution.
Parametrise x+y=2, z=1, 2x+2y=4.
2 marks
The last equation is redundant.
Choose y=t and solve for x.
$(2 - t, t, 1)$ , real t.
Solve x+2y+z=8, y-z=1, x-z=2.
2 marks
Use y=z+1 and x=z+2.
Substitution gives 4z+4=8.
$(3, 2, 1)$
What does a unique solution of three independent plane equations represent?
2 marks
Each equation imposes an independent restriction.
No free direction remains.
Exactly one common point of all three planes.

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Row operations and geometric interpretation of solution sets

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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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