Question: The average of $3v+2$, $5v-4$, and $4v+7$ is 15. What is the value of $v$?
["Understanding How to Solve the Average Problem: Find $ v $ from $ \frac{(3v+2) + (5v-4) + (4v+7)}{3} = 15 $", "When faced with a math problem like "The average of $3v + 2$, $5v - 4$, and $4v + 7$ is 15. What is the value of $v$?", solving it step-by-step using algebraic principles makes finding the answer clear and straightforward. This SEO-optimized article explains the best approach to solve such equations, helping students, educators, and learners master average-related algebra problems efficiently.", "---", "### Solving for $ v $ When the Average Is Given", "Question Recap:\nThe average of three expressions: $3v + 2$, $5v - 4$, and $4v + 7$ equals 15. What is the value of $ v $?", "---", "### Step 1: Write the Average Formula", "The average of three numbers is calculated by adding them together and dividing by 3:", "[\n\frac{(3v + 2) + (5v - 4) + (4v + 7)}{3} = 15\n]", "---", "### Step 2: Combine Like Terms in the Numerator", "Combine all terms involving $ v $, then combine the constant terms:", "- Add $ v $-terms:\n $ 3v + 5v + 4v = 12v $", "- Add constant terms:\n $ 2 - 4 + 7 = 5 $", "So the numerator simplifies to:\n[\n12v + 5\n]", "Now the equation becomes:\n[\n\frac{12v + 5}{3} = 15\n]", "---", "### Step 3: Eliminate the Denominator", "Multiply both sides of the equation by 3 to eliminate the fraction:\n[\n12v + 5 = 45\n]", "---", "### Step 4: Solve for $ v $", "Subtract 5 from both sides:\n[\n12v = 40\n]", "Divide both sides by 12:\n[\nv = \frac{40}{12} = \frac{10}{3}\n]", "---", "### Final Answer", "[\n\boxed{v = \frac{10}{3}}\n]", "---", "### SEO Optimization Tips for This Article", "- Include keywords: solve for v, average of expressions, algebraic average, how to find v from average, step-by-step equation solving\n- Use schema-compatible structure: Question → Steps → Final Answer\n- Add related internal links: “liquid formulas for linear equations,” “algebraic word problems explained”\n- Engage readers with real-world context: “This method helps students quickly solve common school math problems involving averages.”\n- Optimize readability: Short paragraphs, bullet points, bold key steps (e.g., Combine like terms).", "---", "By understanding how to compute the average and isolate the variable step-by-step, learners can confidently tackle similar algebra problems, making this article a valuable resource for students and educators alike. Mastering these techniques improves mathematical fluency and problem-solving speed—key components in academic success."]
