Systems of equations can be solved by a variety of methods. The simplest are:

**Substitution**

In the system of equations \( x+y = 5 \) and \( x-y = 3 \), we can solve for one of the variables and then substitute that relation into the other equation. From the second equation, we see that \( x = y+3 \), so we can rewrite the first equation as \( (y+3) + y = 5 \), thus giving us \( 2y = 2 \implies y = 1 \). Thus \( y = 1 \) and \( x = 4 \).

**Elimination**

Using the same pair of equations as above, we could also combine the two equations together in such a way as to eliminate one of the variables. In this case, we can simply add them together. Thus we have \( (x+y) + (x-y) = 3 + 5 \), which gives us \( 2x = 8 \implies x = 4 \). Thus \( y = 1\) and \( x = 4 \).

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