v_1 + 2(6 + 2v_1) + 3(11 + 3v_1) = 0 \implies v_1 + 12 + 4v_1 + 33 + 9v_1 = 0 \implies 14v_1 + 45 = 0 \implies v_1 = -\frac{45}{14}.

["Understanding the Equation: Solving for ( v_1 ) Step-by-Step", "When tackling complex algebraic equations, simplifying expressions correctly is key to arriving at the right solution. This article walks through a common linear equation involving ( v_1 ), showing how to simplify and solve it step by step.", "---", "### The Equation", "We begin with the equation:", "[\nv_1 + 2(6 + 2v_1) + 3(11 + 3v_1) = 0\n]", "---", "### Step 1: Expand Parentheses", "The equation contains parentheses with expressions inside. Apply the distributive property to expand each term:", "- Expand ( 2(6 + 2v_1) ):\n [\n 2 \cdot 6 + 2 \cdot 2v_1 = 12 + 4v_1\n ]", "- Expand ( 3(11 + 3v_1) ):\n [\n 3 \cdot 11 + 3 \cdot 3v_1 = 33 + 9v_1\n ]", "Substitute back into the original equation:", "[\nv_1 + (12 + 4v_1) + (33 + 9v_1) = 0\n]", "---", "### Step 2: Remove Parentheses and Combine Like Terms", "Now remove parentheses and group like terms (terms with ( v_1 ) and constant numbers):", "[\nv_1 + 12 + 4v_1 + 33 + 9v_1 = 0\n]", "Combine all ( v_1 ) terms:\n[\nv_1 + 4v_1 + 9v_1 = (1 + 4 + 9)v_1 = 14v_1\n]", "Combine constants:\n[\n12 + 33 = 45\n]", "Resulting equation:", "[\n14v_1 + 45 = 0\n]", "---", "### Step 3: Solve for ( v_1 )", "Isolate ( 14v_1 ):", "[\n14v_1 = -45\n]", "Divide both sides by 14:", "[\nv_1 = -\frac{45}{14}\n]", "---", "### Final Answer:", "[\n\boxed{v_1 = -\frac{45}{14}}\n]", "---", "### Why This Matters", "Understanding how each step—distributing, combining, and solving—transforms a complex equation into a clear, solvable form is essential in algebra. This method applies broadly to linear equations and supports deeper problem-solving skills in mathematics, physics, engineering, and computer science.", "By methodically simplifying expressions, you build confidence in handling equations of greater complexity, unlocking powerful analytical tools used in advanced mathematics and real-world modeling.", "---", "Keywords for SEO:\n- Solve linear equation\n- Algebraic simplification\n- Solve for ( v_1 )\n- Step-by-step equation solving\n- Mathematical problem solving\n- Linear equation steps\n- Solve ( v_1 + 2(6 + 2v_1) + 3(11 + 3v_1) = 0 )", "---", "Understanding such equations clearly ensures accuracy and reinforces foundational algebra skills critical across STEM fields."]