The first step is to make sure the equations are setup such that either the x terms or the y terms are opposites of each other (here, +13y and -13y satisfy this condition, but in general you may need to multiply an entire equation by a certain number to make this happen). Next, add one equation to the other like so:
4x + 13y = 16
2x - 13y = 8
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6x + 0y = 24
Note, the coefficient on the y term is zero (a result of having the opposites set up). This is important because we went from a system of two equations with two unknowns, to one equation with one unknown, namely, 6x = 24)
Solve 6x=24 and get x=4. Now plug x=4 into one of the original equations and find the y value.
2x - 13y = 8 ==> 2(4) - 13y = 8 ==> 8 - 13y = 8 ==> -13y = 0 ==> y=0
The solution to the system of equations is x=4, y=0.
Check this solution by plugging both x=4 and y=0 into both equations, and verify you get true statements.
4x + 13y = 16 ==> 4(4) + 13(0) = 16 + 0 = 16 True
2x - 13y = 8 ==> 2(4) - 13(0) = 8 - 0 = 8 True