Question 245529
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The Linear Combinations Method is also known as the Addition Method which is also known as the Elimination Method.  I like to call it the Elimination Method because that name most aptly describes what you are trying to accomplish, namely the elimination of one of the variables so that you can easily solve for the other.

The first step of the method is to multiply one or the other (or sometimes both) of the equations by a number or numbers so that the coefficients on one of the variables becomes the additive inverse of the coefficient on that same variable in the other equation.

In your very simple example, you don't have to do the first step -- it is already done for you with respect to the variable *[tex \Large y].  Note that the coefficient on *[tex \Large y] in the first equation is 1 and the coefficient on *[tex \Large y] in the second equation is -1.  1 and -1 are additive inverses because 1 + (-1) = 0.

The next step is to add the two equations together (that's where the Addition Method name comes from).  Just like any other simplification, you combine like terms.

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ y\ =\ 4]

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \,x\ -\ y\ =\ 2]

*[tex \Large 2x\ +\ x\ =\ 3x], *[tex \Large y\ +\ (-y)\ =\ 0y], and *[tex \Large 4\ +\ 2\ = 6], that is:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ +\ 0y\ =\ 6]

Notice that we now have a zero coefficient on *[tex \Large y], so we can just eliminate (there's that name again) the *[tex \Large y] variable and simply write:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ =\ 6]

Which solves to:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \,x\ =\ 2]

Now that you have a value for *[tex \Large x], it is a simple matter to substitute that value back into one of the original equations and solve for the other variable.

However, I'm going to do this the hard way as a demonstration of the step 1 that you didn't have to do.  Let's eliminate the *[tex \Large x] variable instead.

Begin with the given equations:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ y\ =\ 4]

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \,x\ -\ y\ =\ 2]

Multiply the second equation by -2:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ \ 2x\ +\ y\ \ \,=\ \ \,4]

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -2x\ +\ 2y\ =\ -4]

Now the coefficients on *[tex \Large x] are additive inverses, that is 2 + (-2) = 0.  Add the equations:

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 0x\ +\ 3y\ =\ 0]

and finally

*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 0]

That's all there is to it.

John
*[tex \LARGE e^{i\pi} + 1 = 0]
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