-3(-1 - 2v_2) + 2v_3 = 5 \implies 3 + 6v_2 + 2v_3 = 5 \implies 6v_2 + 2v_3 = 2 \implies 3v_2 + v_3 = 1

["Understanding and Solving the Linear Equation: –3(–1 – 2v₂) + 2v₃ = 5", "When tackling linear equations in multiple variables, simplifying and solving step-by-step is key to clarity and accuracy. This article explores the equation:", "–3(–1 – 2v₂) + 2v₃ = 5", "and demonstrates how it simplifies to a clear, solvable form:\n3v₂ + v₃ = 1", "---", "### Step 1: Expand the Expression Inside Parentheses", "Start by carefully distributing the coefficient –3 across the parentheses (–1 – 2v₂):", "[\n-3 \cdot (-1) + (-3) \cdot (-2v_2) = 3 + 6v_2\n]", "So the equation becomes:\n[\n3 + 6v_2 + 2v_3 = 5\n]", "---", "### Step 2: Isolate the Constants", "Subtract 3 from both sides to simplify the constant terms:", "[\n6v_2 + 2v_3 = 5 - 3\n]\n[\n6v_2 + 2v_3 = 2\n]", "---", "### Step 3: Simplify by Factoring", "Factor the left-hand side by pulling out the common factor of 2:", "[\n2(3v_2 + v_3) = 2\n]", "Divide both sides by 2:", "[\n3v_2 + v_3 = 1\n]", "---", "### Summary", "The original equation:\n–3(–1 – 2v₂) + 2v₃ = 5\nsimplifies step-by-step to:\n[\n\boxed{3v_2 + v_3 = 1}\n]", "This simplified form is easier to analyze, solve for one variable, or substitute into larger systems. Understanding each step ensures clarity in solving linear equations involving multiple variables.", "---", "### Why This Equation Matters", "Linear equations like this are foundational in fields such as engineering, economics, and computer science. Simplified forms help identify relationships between variables and assist in graphing, optimization, and modeling real-world scenarios.", "---", "### Conclusion", "By systematically expanding, simplifying, and factoring, complex equations become manageable. Remember:\n1. Distribute carefully\n2. Combine like terms\n3. Factor when possible", "This approach turns abstract expressions into clean, usable results — essential for both hand solutions and computational algorithms.", "---", "Keywords: linear equation, solve v₂ and v₃, simplify 3v₂ + v₃ = 1, algebraic simplification, step-by-step equation solving, list v₂ and v₃ – 3(-1 − 2v₂) + 2v₃ = 5."]