The aim of the present work is to consider a mixed boundary value problem of the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges $y = 0$, $y = h$ having clamped conditions on intervals $|x| \leq 6$ a and hinged support conditions for $|x| > a$. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of a solution of the system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the system of dual integral equations to an infinite system of linear algebraic equations is also proposed.