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 Context

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

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

Key Insights

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:
$$
rac{9y + 6}{3} = 10
$$

Continue Reading

So 39 is valid and the largest.
Hence, the largest possible total duration is $ oxed{39} $.
Question: A deep learning model processes patient data in batches, each batch size a multiple of 11, and the total dataset size is a multiple of 9. If the model must process at least 100 but fewer than 200 patients, what is the smallest possible batch size (in patients) that allows full utilization of both constraints?

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Final Thoughts

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
$$
oxed{ rac{8}{3}}
$$