Mathematical mistakes are interesting

Algebra Level 3

Professor Ávila gave Esmeralda an equation of the form \(ax = b\), where \(a\) and \(b\) are real numbers. Esmeralda, by accident, solved the equation \(bx = a\) and got a solution that is equal to the correct one minus 60. If the correct solution is of the form \(m + \sqrt{n}\), where \(m\) and \(n\) are integers, what is the value of \(m + n\)?


Source: Brazilian Maths Olympiad (OBM), 2015.
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