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The La Silla/QUEST Variability Survey (LSQ) and the Carnegie Supernova Project (CSP II) are collaborating to discover and obtain photometric light curves for a large sample of low-redshift (z< 0.1) Type Ia supernovae (SNe Ia). The supernovae are discovered in the LSQ survey using the 1 m ESO Schmidt telescope at the La Silla Observatory with the 10 square degree QUEST camera. The follow-up photometric observations are carried out using the 1 m Swope telescope and the 2.5 m du Pont telescopes at the Las Campanas Observatory. This paper describes the survey, discusses the methods of analyzing the data, and presents the light curves for the first 31 SNe Ia obtained in the survey. The SALT 2.4 supernova light-curve fitter was used to analyze the photometric data, and the Hubble diagram for this first sample is presented. The measurement errors for these supernovae averaged 4%, and their intrinsic …