Solution: Average equation: $ \frac{(3y + 1) + (y + 7) + (5y - 2)}{3} = 10 $. Simplify numerator: $ 9y + 6 = 30 $. Subtract 6: $ 9y = 24 $. Divide: $ y = \frac{8}{3} $. \boxed{\dfrac{8}{3}}

["Solving the Average Equation: Step-by-Step Guide to Find ( y = \dfrac{8}{3} )", "Understanding how to solve an average equation is a fundamental skill in algebra, useful for students, educators, and anyone working with data statistics. Today, we’ll walk through a clear and structured solution to the equation:", "[\n\frac{(3y + 1) + (y + 7) + (5y - 2)}{3} = 10\n]\nThis problem demonstrates how to simplify an average expression, simplify the numerator, solve for ( y ), and verify the result.", "---", "### Step 1: Simplify the Numerator", "The equation involves an average of three expressions:\n[\n\frac{(3y + 1) + (y + 7) + (5y - 2)}{3}\n]", "Combine like terms in the numerator:\n- Combine ( y ) terms: ( 3y + y + 5y = 9y )\n- Combine constant terms: ( 1 + 7 - 2 = 6 )", "So, the numerator simplifies to:\n[\n9y + 6\n]", "---", "### Step 2: Set Up the Simplified Equation", "Now substitute the simplified numerator back into the original equation:\n[\n\frac{9y + 6}{3} = 10\n]", "Multiply both sides by 3 to eliminate the denominator:\n[\n9y + 6 = 30\n]", "---", "### Step 3: Solve for ( y )", "Subtract 6 from both sides:\n[\n9y = 24\n]", "Divide both sides by 9:\n[\ny = \frac{24}{9} = \frac{8}{3}\n]", "---", "### Final Answer", "[\n\boxed{y = \dfrac{8}{3}}\n]", "---", "### Why This Matters", "Solving average equations helps build pattern recognition in algebra and prepares learners for real-world applications like calculating averages, optimizing data, or interpreting statistical results. Mastering each step ensures a stronger foundation in mathematical reasoning.", "If you want to deepen your skills, practice similar equations and explore how changing coefficients or constants affect solutions — frequent practice enhances fluency and confidence!"]