Explanation:
are you sure you gave us the correct equations ? no typos, nothing left out ?
just to be clear, what you wrote was
2x/2 - 5y/3 = -2
x/3 + y/2 = 13/6
"elimination" means that we multiply both equations by fitting factors and then add them up, so that in the result we get one equation with only one variable (the second variable gets "eliminated"). after solving this, we can use that result in one of the original equations to get the solution for the second variable.
as it looks like we are dealing in both equations with fractions of 1/2 and 1/3 and also 1/6. the typical common denominator for this is 1/6.
so, first, let's try and bring everything to multiples of 1/6.
6x/6 - 10y/6 = -12/6
2x/6 + 3y/6 = 13/6
now we see, we can multiply the second equation by -3, and then we can sum both equations eliminating x :
6x/6 - 10y/6 = -12/6
-6x/6 + -9y/6 = -39/6
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0 -19y/6 = -51/6
after multiplying both sides by -6 :
19y = 51
y = 51/19 = 2.684210526...
using e.g. the first original equation
2x/2 - 5y/3 = -2
x - 5y/3 = -2
x - 5(51/19)/3 = -2
x - 5×51/(19×3) = -2
x - 255/57 = -2
x = -2 + 255/57 = -114/57 + 255/57 = 141/57 = 47/19 =
= 2.473684211...