Definition:Dependent Equations
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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
- 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