What is the purpose of using the elimination method?

The elimination method reduces the problem to solving a one variable equation.
For example, look at the following system of two variables:
##2x+3y=1##
##-2x+y=7##
It is relatively difficult to determine the values of ##x## and ##y## without manipulating the equations. If one adds the two equations together, the ##x##s cancel out; the ##x## is eliminated from the problem. Hence it is called the “elimination method.”
One ends up with:
##4y=8##
From there, it is trivial to find ##y##, and one can simply plug the value of ##y## back into either equation to find ##x##.

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