Question: The average of $4y - 3$, $2y + 5$, and $y + 7$ is 10. Find $y$.
["# How to Find $ y $ When the Average of $ 4y - 3 $, $ 2y + 5 $, and $ y + 7 $ Is 10", "Understanding averages is a foundational math concept, but applying it to real expressions can help sharpen problem-solving skills. In this article, we’ll break down how to find the value of $ y $ when the average of three algebraic expressions equals 10. Specifically, we’ll solve the equation based on the average of $ 4y - 3 $, $ 2y + 5 $, and $ y + 7 $, which is given as 10.", "## What Is the Average of Three Numbers?", "The average (or mean) of three numbers is found by summing them and dividing by 3. For algebraic expressions, the same principle applies:\n$$\n\ ext{Average} = \frac{(4y - 3) + (2y + 5) + (y + 7)}{3}\n$$\nWe are told this average equals 10.", "## Step 1: Add the Three Expressions", "Combine like terms in the numerator:\n$$\n(4y - 3) + (2y + 5) + (y + 7) = 4y + 2y + y + (-3 + 5 + 7) = 7y + 9\n$$", "## Step 2: Set Up the Equation", "Now substitute the sum back into the average formula:\n$$\n\frac{7y + 9}{3} = 10\n$$", "## Step 3: Solve for $ y $", "Multiply both sides by 3 to eliminate the denominator:\n$$\n7y + 9 = 30\n$$", "Subtract 9 from both sides:\n$$\n7y = 21\n$$", "Divide both sides by 7:\n$$\ny = 3\n$$", "## Why This Method Works", "This technique uses algebra to translate word problems into equations, then applies basic algebraic operations—combining like terms, distributing, and solving linear equations. It’s a reliable method applicable to many real-world and academic math challenges.", "## Final Answer", "The value of $ y $ that makes the average of $ 4y - 3 $, $ 2y + 5 $, and $ y + 7 $ equal to 10 is:", "$ y = 3 $", "---", "Keywords: average of expressions, solve for $ y $, linear equations, algebra practice, math homework help, solve algebraic average, basics of mean in algebra, step-by-step algebra, find $ y $, $ 4y - 3 $, $ 2y + 5 $, $ y + 7 $, real-world math problems", "---", "By mastering such problems, students build confidence in algebra and prepare for more advanced topics in mathematics. Whether studying for a test or solving on your own, always verify your solution by plugging $ y = 3 $ back into the original expressions and confirming the average indeed equals 10."]
