Solving systems of 2 linear equations in 2 variables by Gaussian Elimination
In the Gaussian Elimination method
you eliminate variables by transforming the system of equations into row-echelon form by means of row operations. Then the system is solved by back-substitution.
Let's solve the following system of equations using Gaussian elimination
Divide the first equation by 3
Multiply (*) by 4 and add -1 times to the second equation. We get the following system
From the second equation y=2. Substitution this into the first equation gives x
Solving a system of two linear equations in two variables with substitution method
The main idea of the substitution method is to solve one of the variables in terms of the others (it does not matter which equation we choose) and then substitute the result into another equation.
Consider the following system:
Solving the second equation for y gives y = 4x - 14, and substituting this into the first equation gives x = (16-2(4x-14))/3.
Solving the first equation we get x = 4.
Substituting x = 4 into the second equation gives y = 2.