Question: The average of $ 3y + 1 $, $ y + 7 $, and $ 5y - 2 $ is 10. Solve for $ y $.

February 22, 2026

Title: How to Solve for $ y $ When the Average of $ 3y + 1 $, $ y + 7 $, and $ 5y - 2 $ Is 10

When solving mathematical word problems in algebra, one common task is finding the value of a variable when the average of given expressions is known. In this article, we’ll break down how to find $ y $ when the average of $ 3y + 1 $, $ y + 7 $, and $ 5y - 2 $ equals 10.

Understanding the Problem

We’re told that the average of three expressions — $ 3y + 1 $, $ y + 7 $, and $ 5y - 2 $ — is 10. To find $ y $, we start by remembering that the average of three numbers is the sum divided by 3.

So the equation becomes: $$ rac{(3y + 1) + (y + 7) + (5y - 2)}{3} = 10 $$

Step 1: Combine Like Terms in the Numerator

Let’s simplify the expression inside the parentheses: $$ (3y + 1) + (y + 7) + (5y - 2) $$

Add the $ y $-terms: $ 3y + y + 5y = 9y $

Add the constant terms: $ 1 + 7 - 2 = 6 $

So the sum is $ 9y + 6 $. Now substitute back into the equation: $$ rac{9y + 6}{3} = 10 $$

Step 2: Simplify the Fraction

Divide numerator by 3: $$ 3y + 2 = 10 $$

Step 3: Solve for $ y $

Subtract 2 from both sides: $$ 3y = 8 $$

Divide both sides by 3: $$ y = rac{8}{3} $$

Final Answer

The value of $ y $ that makes the average of $ 3y + 1 $, $ y + 7 $, and $ 5y - 2 $ equal to 10 is $$ oxed{rac{8}{3}} $$

Why This Matters for Students and Learners

Understanding how to compute averages and solve linear equations is essential in algebra. This type of problem teaches vital skills like combining expressions, simplifying fractions, and translating real-world averages into mathematical equations. Mastering these steps lays a strong foundation for more advanced math topics.

Keywords: solve for $ y $, average of expressions, algebra problem, equation solving, linear equation, $ 3y + 1 $, $ y + 7 $, $ 5y - 2 $, average equals 10, step-by-step math, math tutorial

Meta Description: Learn how to solve for $ y $ when the average of $ 3y + 1 $, $ y + 7 $, and $ 5y - 2 $ is 10. Step-by-step guide with clear algebra explanation and final answer $ y = rac{8}{3} $. Perfect for students mastering equations and averages.