Definition:Elimination
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Definition
Elimination is the process of solving a system of simultaneous equations by removing variables by algebra.
Examples
Arbitrary Example $1$
Consider the system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x + 3 y\) | \(=\) | \(\ds 7\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x + y\) | \(=\) | \(\ds 9\) |
This can be solved by multiplying $(2)$ by $3$ and then subtracting $(1)$ from the resulting equation:
\(\text {(3)}: \quad\) | \(\ds 6 x + 3 y\) | \(=\) | \(\ds 27\) | $(2) \times 3$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 5 x\) | \(=\) | \(\ds 20\) | $(3) - (1)$ |
Hence $y$ has been eliminated, and we have:
- $x = 4$
from which we can substitute for $x$ in either $(1)$ or $(2)$ and obtain $y = 1$.
Arbitrary Example $2$
Consider the system of simultaneous linear equations:
\(\text {(1)}: \quad\) | \(\ds x + 3 y\) | \(=\) | \(\ds 7\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds 2 x + y\) | \(=\) | \(\ds 9\) |
This can be solved by putting $(1)$ in the form:
- $x = 7 - 3 y$
from which we can substitute for $x$ in either $(2)$ to obtain:
- $2 \paren {7 - 3 y} + y = 9$
from which it follows that $y = 1$ and $x = 4$.
Also see
- Results about elimination can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): elimination
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elimination