Proof for mathd_algebra_513 - #62
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Proven lemmas: 1/1
The goal is to prove that if real numbers a and b satisfy 3a + 2b = 5 and a + b = 2, then necessarily a = 1 and b = 1. This is a straightforward linear algebra problem over ℝ: the two equations should determine the two unknowns uniquely.
The proof was decomposed exactly along the conclusion a = 1 ∧ b = 1, so there are two sub-goals: prove a = 1 and prove b = 1. Both of these were solved directly from the hypotheses using linear arithmetic reasoning.
Progress is complete: 1 theorem out of 1 has been proved, and both sub-goals are finished. There is nothing remaining to be worked on.
The key strategy was to use the two linear equations together and let Lean’s linear-arithmetic tactic derive each variable value automatically. Mathematically, this corresponds to solving the small system of equations, and no extra lemmas or case splits were needed.