This paper introduces a novel constraint learning mechanism for Constraint Programming (CP) solvers that integrates cutting planes reasoning into the conflict analysis procedure. Drawing inspiration from Lazy Clause Generation (LCG), our approach, named Lazy Linear Generation (LLG), can generate linear integer inequalities to prune the search space, rather than propositional clauses as in LCG. This combines the strengths of constraint programming (strong propagation through global constraints) with cutting-planes reasoning. We present linear constraint explanations for various arithmetic constraints and the element constraint. An experimental evaluation shows that the improved generality of linear constraints has a practical impact on a CP solver by reducing the number of encountered conflicts in 45% of our benchmark instances. Our analysis and prototype implementation show promising results and are an important step towards a new paradigm to make constraint programming solvers more effective.