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(1) The substitution method. This method is best utilized when one of the variables in one of the equations has a coefficient of 1 or -1, otherwise you will introduce fractions. Substitution can also be used for nonlinear systems of equations.
(2) Linear combinations (also called the elimination method, multiplication and addition method, etc...) My personal favorite as it can be done efficiently. It generalizes well to larger systems and is the underpinning of various other solution methods that use matrices.
As the name implies it requires the equations to be linear.
You need to know both and be comfortable switching between them.
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