Definition:Dependent Equations

Definition

An equation is dependent on a set of simultaneous equations if and only if it is satisfied by every set of values of the variables that satisfies the set of equations.

A set of simultaneous equations is dependent if and only if one of them is dependent on the others.

Examples

Arbitrary Example

Consider the set of simultaneous equations:

\(\text {(1)}: \quad\)

\(\ds x + y\)

\(=\)

\(\ds 3\)

\(\text {(2)}: \quad\)

\(\ds x \paren {x + y}\)

\(=\)

\(\ds 3 x\)

Equations $(1)$ and $(2)$ are dependent because every $\tuple {x, y}$ which satisfies $(1)$ also satisfies $(2)$.

Also see

• Definition:Independent Equations

• Results about dependent equations can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): dependent equations
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): dependent equations