Definition:Eliminant

Definition

Let $S$ be a system of simultaneous linear equations.

The eliminant of $S$ is the determinant formed by removing the variables between the equations.

Examples

Arbitrary Example

Consider the system of simultaneous linear equations:

\(\ds a_1 x + b_1 y + c_1\)

\(=\)

\(\ds 0\)

\(\ds a_2 x + b_2 y + c_2\)

\(=\)

\(\ds 0\)

\(\ds a_3 x + b_3 y + c_3\)

\(=\)

\(\ds 0\)

The eliminant of the above is:

$\begin {vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end {vmatrix}$

Also known as

An eliminant is also known as a resultant.

Also see

• Eliminant of Consistent Simultaneous Linear Equations is Zero

• Results about eliminants can be found here.

Sources

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