rac{(a - 1)(a - 3)}{a - 1} = 2 \implies a - 3 = 2 \quad (a

["Title: Solving the Equation (a − 1)(a − 3)/(a − 1) = 2: Step-by-Step Explanation with a ↔ 2 Implication", "---", "Introduction\nSolving rational equations is a fundamental skill in algebra, and one common challenge involves simplifying expressions before isolating the variable. In this article, we explore the equation:\n[\n\frac{(a - 1)(a - 3)}{a - 1} = 2 \implies a - 3 = 2 \quad (a <br/>\ne 1)\n]\nOur goal is to clearly explain why and how this simplification occurrs, and why the condition ( a <br/>\ne 1 ) is essential. This guide is perfect for students, math learners, or anyone looking to strengthen their algebraic reasoning around equation simplification and logical implications.", "---", "Understanding the Equation", "We start with:\n[\n\frac{(a - 1)(a - 3)}{a - 1} = 2\n]", "The expression involves a fraction with numerator ((a - 1)(a - 3)) and denominator ((a - 1)). Before canceling, it’s crucial to recognize where the denominator equals zero, since division by zero is undefined.", "### Step 1: Domain restriction\nThe expression is undefined when ( a - 1 = 0 ), i.e.,\n[\na = 1\n]\nThus, ( a <br/>\ne 1 ) is a strict condition. We are solving the equation under the constraint ( a <br/>\ne 1 ).", "---", "Step 2: Simplify the rational expression", "Because ( a <br/>\ne 1 ), the factor ( a - 1 ) in the denominator does not equal zero, and we are safe to cancel it:\n[\n\frac{(a - 1)(a - 3)}{a - 1} = a - 3 \quad \ ext{(for } a <br/>\ne 1\ ext{)}\n]", "Thus, the original equation simplifies to:\n[\na - 3 = 2\n]", "---", "Step 3: Solve the simplified equation", "Now solve linearly:\n[\na - 3 = 2 \quad \Rightarrow \quad a = 5\n]", "---", "Step 4: Verify the solution fits the domain", "Our solution ( a = 5 ) satisfies ( a <br/>\ne 1 ), so it is valid. Plugging back into the original equation:\n[\n\frac{(5 - 1)(5 - 3)}{5 - 1} = \frac{4 \cdot 2}{4} = 2\n]\nWhich confirms the solution is correct.", "---", "Why does ( a - 3 = 2 ) imply this simplification?", "The simplification relies on the fact that ( a <br/>\ne 1 ). Because ( a - 1 ) appears in both the numerator and denominator, we are authorized to cancel it — this is a valid algebraic operation only when the denominator isn’t zero. The implication\n[\n\frac{(a - 1)(a - 3)}{a - 1} = 2 \quad \implies \quad a - 3 = 2 \quad \ ext{(with } a <br/>\ne 1\ ext{)}\n]\nis logically sound under the domain restriction.", "Note: We never truly "solve" ( a - 1 = 0 ); instead, we exclude that value and simplify the expression elsewhere.", "---", "Conclusion", "Simplifying rational expressions often lets us reduce complex equations to linear forms — but only when properly respecting the domain. In this case:\n[\n\frac{(a - 1)(a - 3)}{a - 1} = 2 \implies a - 3 = 2 \quad (a <br/>\ne 1)\n]\nThis lies at the heart of rational equation solving: cancel cautiously, check domain, verify solution.", "Understanding this process helps avoid common pitfalls and strengthens your algebraic toolkit. Whether tackling exams, homework, or real-world problems, mastering simplifications turns tricky equations into manageable steps.", "If you found this explanation useful, share it with fellow learners — and remember: always skip the dangerous steps (like canceling under zero) and double-check your domain!", "---", "Keywords for SEO:\nrational equation simplification, solve (a - 1)(a - 3)/(a - 1) = 2, cancel denominator algebra, domain restriction in equations, a ≠ 1 implication, step-by-step equation solving, linear equation from rational expression, avoid division by zero in algebra", "Meta Description:\nLearn how to simplify (\frac{(a - 1)(a - 3)}{a - 1} = 2) correctly, including why (a <br/>\ne 1) and how it leads to (a - 3 = 2). Step-by-step guide with domain awareness for precise algebra."]