To solve for the values in the variable knowing that the sum of all angles in a triangle add to 180 degrees. Simply use this fact to determine the value of z.
Z = 180 - 115 = 165
Then to solve for x, do the same thing. Add up 51 and 45 and subtract the sum by the 165 degrees.
To solve for y, simply knowing the angle that is present for both 70 and the unknown is 180. Subtract 180 from y.
To actually solve for y add up the value for x and 51 and subtract that amount by 180 degrees.
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From the law of sines, we have:

,
where x and y are the sides opposite to angles X and Y, respectively.
Substituting the known values, we have:

, thus

.
Using a calculator, we can find that arcsin(0.31)=18 degrees, approximately.
We know that sine of (180-18)=162 degrees is also 0.31. But 162 and 51 degrees add up to more than 180 degrees.
Thus, there is only one triangle that can be formed under these conditions.