Definition:Consistent Simultaneous Equations
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Definition
A system of simultaneous equations is referred to as consistent if and only if it has at least one solution.
That is, if and only if there exists a set of values for its variables such that all the equations are satisfied.
Inconsistent
A set of equations is described as inconsistent if and only if they are not consistent
That is, there exists no set of values for its variables such that all the equations are satisfied.
Examples
Arbitrary Example $1$
Consider the simultaneous equations:
| \(\ds x + y\) | \(=\) | \(\ds 10\) | ||||||||||||
| \(\ds x + 2 y\) | \(=\) | \(\ds 15\) |
These are satisfied with the values:
| \(\ds x\) | \(=\) | \(\ds 5\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds 5\) |
and so are consistent.
Arbitrary Example $2$
Consider the simultaneous equations:
| \(\ds x + y\) | \(=\) | \(\ds 10\) | ||||||||||||
| \(\ds x + y\) | \(=\) | \(\ds 15\) |
These cannot be satisfied by any pair of values for $x$ and $y$.
Hence they are inconsistent.
Also see
- Results about consistent simultaneous equations can be found here.
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Definition $1.6$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): consistent: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): consistent: 1.