Underdetermined system of equations

High School

Definition

An equation that consists of more variables than equations. Underdetermined equations either have no solution (inconsistent) or a multiple number of solutions (consistent). For example X + Y + Z = 1, X + Y + Z = 0 has 3 variables and 2 solutions. It does not have any solutions, therefore this underdetermined equation is inconsistent. X + Y + Z = 1, X + Y + 2Z = 3 would be a consistent equation because there can be multiple solutions to this equation.

Worked examples

\(x + y = 3\), \(2x + 2y = 6\)

Two equations, two unknowns, but the second is just double the first — infinitely many solutions lie on that line.

\(x + y + z = 1\), \(x + y + z = 0\)

Three variables, two equations, but they contradict each other — no solution exists (inconsistent).

\(x + y + z = 1\), \(x + y + 2z = 3\)

Three variables, two equations that agree — infinitely many solutions parameterized by one free variable (consistent).

Common mistakes

Fewer equations than variables always means no solution. → Fewer equations than variables can have infinitely many solutions if consistent. Underdetermined systems can be consistent (infinite solutions) or inconsistent (no solution).
\(x + y = 1\), \(2x + 2y = 3\) has infinite solutions because two equations, two variables. → No solution — the equations contradict (inconsistent). The second equation simplifies to \(x + y = 1.5\), which conflicts with the first.
You can always solve for all variables in an underdetermined system. → You can only express some variables in terms of others (parameters). With more variables than equations, at least one variable remains free.

Where you'll use it next

You'll encounter underdetermined systems in linear algebra when studying solution spaces, in optimization with constraints, and in physics when modeling systems with degrees of freedom.

Found in 1 StudyPug lesson

Systems of Linear Equations: From Theory to Practice

Grade 11 Math

Dive into the world of systems of linear equations. Learn powerful solving techniques, understand solution types, and apply your knowledge to real-world scenarios in economics, engineering, and more.

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