Solution: Average equation: $ \frac{(4z - 3) + (2z + 5) + (z + 1)}{3} = 6 $. Simplify numerator: $ 7z + 3 = 18 $. Subtract 3: $ 7z = 15 $. Divide: $ z = \frac{15}{7} $. \boxed{\dfrac{15}{7}}

["Solving the Average Equation: Step-by-Step Guide to Finding $ z = \dfrac{15}{7} $", "Understanding how to solve average equations is a foundational skill in algebra. Whether in academics or real-world problem-solving, average calculations appear frequently in statistics, finance, and data analysis. In this article, we’ll break down the step-by-step solution to the equation:", "$$\n\frac{(4z - 3) + (2z + 5) + (z + 1)}{3} = 6\n$$", "This equation represents the average of three linear expressions set equal to 6. Let’s simplify and solve it clearly.", "---", "### Step 1: Simplify the Numerator", "The numerator consists of three terms: $ 4z - 3 $, $ 2z + 5 $, and $ z + 1 $. Combine them:", "$$\n(4z - 3) + (2z + 5) + (z + 1)\n$$", "Group like terms:", "- Combine $ z $-terms: $ 4z + 2z + z = 7z $\n- Combine constant terms: $ -3 + 5 + 1 = 3 $", "So, the numerator simplifies to:\n$$\n7z + 3\n$$", "Now the equation becomes:\n$$\n\frac{7z + 3}{3} = 6\n$$", "---", "### Step 2: Eliminate the Denominator", "To simplify, multiply both sides of the equation by 3:", "$$\n7z + 3 = 18\n$$", "This step isolates the expression containing $ z $.", "---", "### Step 3: Solve for $ z $", "Subtract 3 from both sides to eliminate the constant:", "$$\n7z = 18 - 3 = 15\n$$", "Then divide both sides by 7:", "$$\nz = \frac{15}{7}\n$$", "---", "### Final Answer", "$$\n\boxed{\dfrac{15}{7}}\n$$", "---", "### Conclusion", "Solving average equations like this one involves carefully simplifying expressions, combining like terms, and isolating the variable through inverse operations. Recognizing this pattern is essential for mastering algebraic reasoning applicable in schoolwork, coding, data science, and beyond. Use this method to confidently tackle similar equations and strengthen your math foundation."]