Systems of Equations
Consistent, Inconsistent and Dependent Systems
What it means when the variables all vanish, how to tell an impossible system from one with infinitely many solutions, and how to write a general solution.
What this page covers
- Recognise an inconsistent system from its algebra
- Recognise a dependent system and write its general solution
- Interpret each case geometrically
- Use a parameter to describe an infinite solution set
The three outcomes
Solving a linear system always ends in one of three ways, and the algebra announces which before any interpretation is needed.
| Final line | Name | Solutions | Geometry (two variables) |
|---|---|---|---|
| A value for each variable | Consistent, independent | Exactly one | The lines cross once |
| A false statement such as 0 = 3 | Inconsistent | None | The lines are parallel and distinct |
| A true statement such as 0 = 0 | Dependent | Infinitely many | The lines coincide |
The key observation is that in the last two cases the variables vanish. What remains is a numeric statement, and whether it is true or false decides everything.
Inconsistent systems
Worked example — the source's case
Solve 2x + 3y = 2 and 4x + 6y = 1.
The source's note is exact: not possible, so no solution — the equations are inconsistent.
Rewriting both in slope-intercept form shows why:
Equal slopes, different intercepts. The lines are parallel and never meet, which is precisely what 'no solution' means.
Coefficients in the same ratio but constants in a different ratio guarantees inconsistency. Here 42 = 63 = 2 but 12 ≠ 2.
Dependent systems
Worked example — the source's case
Solve 2x + 3y = 2 and -4x - 6y = -4.
A true statement, so the equations are dependent and there are infinitely many solutions. The second equation is just -2 times the first and carries no new information.
The source then writes the general solution using a parameter:
Substituting: 2(1 - 32c) + 3c = 2 - 3c + 3c = 2 ✓ for every c. That the parameter cancels is the confirmation.
The parameter is the free variable, and any of the variables can be chosen for the role. Choosing x = c instead would give y = 2 - 2c3 — a different description of the same infinite set.
Three variables: the same three outcomes
With three unknowns each equation is a plane, and the same three cases arise. The source works all of them.
Consistent — a unique solution
For x + y + z = 5, x - 4y - 3z = 11, -2x + 2y + 5z = -30, elimination gives z = -4, y = 2, x = 7.
In the third equation: -2(7) + 2(2) + 5(-4) = -14 + 4 - 20 = -30 ✓. The third equation was used in the elimination, so the source also verifies against the others.
Dependent — the source's variant
Changing the third equation to -2x + 3y + 2z = -16 gives, after elimination,
Adding the last two gives 0 = 0. The source labels it a dependent system and constructs the general solution with z = c:
Geometrically, the three planes share a whole line rather than a single point.
Inconsistent — the source's variant
With a third equation of 5y + 4z = -5 instead, elimination gives
The source writes inconsistent system. No point lies on all three planes.
Reading the outcome from a row-echelon form
Once a system has been reduced to an augmented matrix in row-echelon form, the outcome can be read off the bottom row without further work.
| Bottom row | Means | Outcome |
|---|---|---|
| A leading 1 in every variable column | Every variable is determined | Unique solution |
| [0 0 0 | k] with k ≠ 0 | 0 = k, false | Inconsistent |
| [0 0 0 | 0] | 0 = 0, true | Dependent; one free variable per zero row |
The source's matrix example makes the point directly: a reduction ending with a row [0 0 0 | 4] is annotated inconsistent — no solution, without any back-substitution being attempted.
The number of free variables equals the number of unknowns minus the number of non-zero rows. One free variable gives a line of solutions; two give a plane.
Common mistakes
| Mistake | Correct | Why |
|---|---|---|
| Treating 0 = 0 as no solution | It means infinitely many | A true statement imposes no restriction |
| Treating 0 = 3 as x = 3 | It means no solution | The variables have already vanished |
| Giving one solution for a dependent system | Give the general solution | Introduce a parameter |
| Concluding an error has been made when variables vanish | It is a legitimate outcome | Two of the three cases do this |
| Not checking the parameter cancels | Substitute the general solution back | It must hold for every value of c |
| Assuming more equations means fewer solutions | Extra dependent equations change nothing | Three concurrent lines still meet at one point |
Frequently asked questions
What does 0 = 3 mean?
That the system is inconsistent and has no solution. The variables have cancelled and left a false statement, so no assignment of values can satisfy all the equations.
What does 0 = 0 mean?
That the equations were not independent — one carries no information the others did not. The system is dependent and has infinitely many solutions.
How do I write infinitely many solutions?
Introduce a parameter. Set the free variable equal to c, express the others in terms of it, and state the solution as an ordered pair or triple involving c.
Can a system with more equations than unknowns be consistent?
Yes, if the extra equations are dependent on the others. Three concurrent lines give a consistent system of three equations in two unknowns.
Source. Handwritten teaching notes, Week 10 pages 10-12 and Week 11 pages 1-3.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.