Question: The average of $ 4x+1 $, $ x+7 $, and $ 2x+4 $ is $ 5x - 2 $. Solve for $ x $.
["Title: How to Solve for $ x $ When the Average of Three Expressions Equals a Linear Function", "Meta Description: Learn how to find the value of $ x $ when the average of $ 4x+1 $, $ x+7 $, and $ 2x+4 $ equals $ 5x - 2 $. Step-by-step solution with explanation.", "---", "Understanding the Average Equation: The Average of $ 4x+1 $, $ x+7 $, and $ 2x+4 $ is $ 5x - 2 $", "Mathematics often presents real-world problems in the form of equations — and one common question is: What value of $ x $ makes the average of three expressions equal a given linear expression? Today, we’ll solve the specific problem:", "> The average of $ 4x+1 $, $ x+7 $, and $ 2x+4 $ is $ 5x - 2 $. Solve for $ x $.", "This type of problem helps build foundational algebra skills and is essential for mastering equation solving and simplifying expressions.", "---", "Step 1: Write the average of the three expressions", "The average of three numbers (or expressions) is the sum divided by 3. So, compute:", "$$\n\frac{(4x + 1) + (x + 7) + (2x + 4)}{3}\n$$", "Step 2: Combine like terms in the numerator", "Add the terms involving $ x $ and constants:", "- Coefficients of $ x $: $ 4x + x + 2x = 7x $\n- Constant terms: $ 1 + 7 + 4 = 12 $", "So the sum is:", "$$\n\frac{7x + 12}{3}\n$$", "---", "Step 3: Set the average equal to $ 5x - 2 $", "According to the problem:", "$$\n\frac{7x + 12}{3} = 5x - 2\n$$", "---", "Step 4: Eliminate the denominator by multiplying both sides by 3", "$$\n3 \cdot \left( \frac{7x + 12}{3} \right) = 3 \cdot (5x - 2)\n$$", "$$\n7x + 12 = 15x - 6\n$$", "---", "Step 5: Solve for $ x $", "Move all $ x $-terms to one side and constants to the other:", "Subtract $ 7x $ from both sides:", "$$\n12 = 15x - 7x - 6\n$$", "$$\n12 = 8x - 6\n$$", "Add 6 to both sides:", "$$\n18 = 8x\n$$", "Divide both sides by 8:", "$$\nx = \frac{18}{8} = \frac{9}{4}\n$$", "---", "Final Answer:\n$$\n\boxed{x = \frac{9}{4}}\n$$", "---", "Why This Equation Matters\nSolving for $ x $ in average equations reinforces the ability to manipulate algebraic expressions, simplify complex fractions, and compare linear functions. It’s a common setup in word problems, science modeling, and economics where averages represent averages of changing quantities.", "---", "Keywords for SEO:\naverage of algebraic expressions, solve for x, linear equations, algebra problem solver, step-by-step average equation, average equals linear expression, solve $ \frac{4x+1 + x+7 + 2x+4}{3} = 5x - 2 $, algebra tutorial, roadmap to solving linear equations", "---", "Stay tuned for more clear, practical algebra guides helping you conquer equations with confidence!"]
