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

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 desired  modulus which is \(14\cdot19=266\).

Thus, data-equation="N=105">\(N=105\) ( data-equation="N=105">\(N=105\) (mod   class="ltx_Math" contenteditable="false" data-equation="mod">\(mod\) \(266\)).