When you use the linear combination method, how can you tell whether you have to multiply by a constant first?

The point of the "linear combination method" is to eliminate one of the variables. If a variable can be eliminated without multiplying by one or more constants, then you do not have to multiply by a constant first.

For example, given the system ...

We can eliminate y simply by adding the equations. We can eliminate x by subtracting one equation from the other. (You might say you need to multiply one of the equations by -1 before you add them in order to eliminate x.)

Now, consider the system ...

Neither variable can be eliminated by simply adding or subtracting one equation to/from the other. One of the equations must be multiplied by -2 or -1/2 before being added to the other to eliminate a variable.

Coefficients in one equation are neither equal to nor opposite the corresponding coefficients in the other equation, so multiplication by a constant is necessary.

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