# Are there more than one way to solve systems of equations by elimination?

Jan 14, 2015

There are more than one way to solve the system of equations

The most utilised methods are elimination and substitution methods. I prefer substitution than elimination.

Other methods like Cramer's rule and other matrix methods such as Gauss elimination, Gauss - Jacobi are available. These are pretty advanced and can solve any number of linear equations.

A comparison of substitution and elimination methods is given below.

Example

$6 x + 4 y = 2$---------->Eqn 1
$x - 2 y = 3$---------->Eqn 2

Elimination method
Multiply Eqn 2 by '2' an add with Eqn 1.

$6 x + 4 y = 2$
$2 x - 4 y = 6$
______+
$8 x = 8$
$x = 1$

Substitute in one of the equations. Using Eqn 1 we have

$6 \cdot 1 + 4 y = 2$
$4 y = 2 - 6$
$y = - 1$

Hence the solution is $x = 1 , y = - 1$

Substitution Method
From Eqn 2 we have

$x = 3 + 2 y$ --> Eqn 3

Substitute in Eqn 1
$6 \cdot \left(3 + 2 y\right) + 4 y = 2$
$18 + 12 y + 4 y = 2$
$16 y = 2 - 18$
$16 y = - 16$
$y = - 1$
Use in eqn 3
$x = 3 + 2. - 1$
$x = 1$
So we get $x = 1 , y = - 1$.