Solving the following equations : $x^{2}+y^{2}-z^{2}=21,\ 3xz+3yz-2xy=18,\ x+y-z=5.$ we get $x=4,3,\dfrac { 2\pm \sqrt { 151 } }{ 3 } ;y=3,4,\dfrac { 2\mp \sqrt { 151 } }{ 3 }$ and $z=2,-\dfrac { 11 }{ 3 }$.

We can verify the solutions by substituting them back into the original equations. Taking the first set of values (x, y, z) = (4, 3, 2), we test the first equation to get 4^2 + 3^2 - 2^2 = 16 + 9 - 4 = 21, which holds true. Testing the second equation yields 3(4)(2) + 3(3)(2) - 2(4)(3) = 24 + 18 - 24 = 18, which is correct. Testing the third equation gives 4 + 3 - 2 = 5, which also matches exactly. Since these values satisfy all three given equations, the statement is True.

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