First, notice that \(\left(14,19\right)=1\). We can find a solution to data-equation="14x+19y=1">\(14x+19y=1\).

By playing around with different numbers, we get our solutions to be \(x=-4\) and \(y=3\).

We now have all our values: \(m_1=14,m_2=19,a_1=3,a_2=7,x=-3,y=7\) and can now solve \(N=m_1a_2x+m_2a_1y\).

     \(=-294+399\)

     \(=105\)

So \(N=105\), however, our unique solution is up to our desried modulus which is \(14\cdot19=266\).

Thus, \(N=105\) (\(mod\) \(266\)).