Question: The average of \( 2v + 4 \), \( 5v + 1 \), and \( 3v + 7 \) is 24. What is the value of \( v \)?
["Understanding the Average: How to Solve for ( v ) with the Expression Average Equal to 24", "Learning how to find the average of expressions is a fundamental algebra skill, especially when solving real-world or mathematical problems involving variables. Today, we explore a common question: The average of ( 2v + 4 ), ( 5v + 1 ), and ( 3v + 7 ) is 24. What is the value of ( v )?", "---", "### Step 1: What Is an Average?", "The average of three numbers is the sum divided by 3. So, for any three expressions, the average is:", "[\n\ ext{Average} = \frac{(2v + 4) + (5v + 1) + (3v + 7)}{3}\n]", "We are told this average equals 24.", "---", "### Step 2: Set Up the Equation", "Substitute the expressions into the average formula:", "[\n\frac{(2v + 4) + (5v + 1) + (3v + 7)}{3} = 24\n]", "Simplify the numerator by combining like terms:", "[\n2v + 5v + 3v = 10v\n\quad \ ext{and} \quad\n4 + 1 + 7 = 12\n]", "So the equation becomes:", "[\n\frac{10v + 12}{3} = 24\n]", "---", "### Step 3: Eliminate the Denominator", "Multiply both sides of the equation by 3 to eliminate the fraction:", "[\n10v + 12 = 72\n]", "---", "### Step 4: Solve for ( v )", "Subtract 12 from both sides:", "[\n10v = 60\n]", "Now divide both sides by 10:", "[\nv = 6\n]", "---", "### Step 5: Verify the Solution", "Plug ( v = 6 ) back into the original expressions:", "- ( 2v + 4 = 2(6) + 4 = 12 + 4 = 16 )\n- ( 5v + 1 = 5(6) + 1 = 30 + 1 = 31 )\n- ( 3v + 7 = 3(6) + 7 = 18 + 7 = 25 )", "Find the average:", "[\n\frac{16 + 31 + 25}{3} = \frac{72}{3} = 24\n]", "Confirmed: the average is indeed 24 when ( v = 6 ).", "---", "### Why This Skill Matters", "Understanding averages helps in math, finance, science, and daily life. Solving for unknowns like ( v ) builds critical thinking and algebraic fluency. Whether balancing budgets, analyzing data, or following equations, mastering averages is essential.", "---", "### Final Answer", "[\n\boxed{6}\n]", "Summary:\nFor the average of ( 2v + 4 ), ( 5v + 1 ), and ( 3v + 7 ) to be 24, the value of ( v ) is 6. Set up the equation, simplify, isolate ( v ), and verify your solution to ensure accuracy.", "---", "Keywords: value of v, solving average problem, algebra basics, equation solving, average of expressions, algebra tutoring, middle school math, high school algebra, solve for v, math education."]
