Solution: First, calculate the sum of the expressions: $(3u - 4) + (7u + 2) + (4u - 1) = 14u - 3$. Divide by 3 to find the average: $\frac{14u - 3}{3}$. Since $u$ is a positive multiple of 3 and $u^2 < 100$, possible values for $u$ are 3, 6. Testing $u = 3$: $\frac{14(3) - 3}{3} = \frac{42 - 3}{3} = \frac{39}{3} = 13$. For $u = 6$, $u^2 = 36 < 100$, but $14(6) - 3 = 81$, $\frac{81}{3} = 27$. However, the problem implies a unique answer, so the smallest valid $u = 3$ gives $\boxed{13}$.

["Solved: Find the Average of a Linear Expression Involving $u$, Given Constraints", "In algebra, simplifying expressions and calculating averages envolves careful computation. A recent problem illustrates how to work with expressions involving a variable $u$, apply constraints, and arrive at a unique solution. Here’s a clear, SEO-optimized breakdown of the solution.", "---", "### How to Calculate the Average of a Linear Expression with Constraints on $u$", "In algebra, when tasked with finding the average of an expression like $(3u - 4) + (7u + 2) + (4u - 1)$, the process begins by combining like terms:", "Step 1: Sum the Expressions\nAdd the three terms algebraically:\n[\n(3u - 4) + (7u + 2) + (4u - 1) = (3u + 7u + 4u) + (-4 + 2 - 1) = 14u - 3\n]", "Step 2: Divide by 3 to Find the Average\nSince the average of three numbers (or expressions) is the sum divided by 3, we compute:\n[\n\frac{14u - 3}{3}\n]", "Step 3: Apply Given Constraints on $u$\nThe problem specifies two key constraints:\n- $u$ is a positive multiple of 3, meaning $u = 3, 6, 9, \dots$\n- $u^2 < 100$ — this limits possible values because $u^2 = 9, 36, 81, \dots$, but $u = 12$ gives $144 > 100$", "So valid values are $u = 3, 6, 9$ (since $9^2 = 81 < 100$, but $12^2 = 144$ exceeds the limit).", "Step 4: Test Each Valid $u$ and Interpret the Problem\nThe problem states that the final result must be unique. Testing $u = 3$:\n[\n\frac{14(3) - 3}{3} = \frac{42 - 3}{3} = \frac{39}{3} = 13\n]\nFor $u = 6$:\n[\n\frac{14(6) - 3}{3} = \frac{81}{3} = 27\n]\nAlthough both satisfy the algebraic steps, the condition of a unique answer implies the smallest valid value of $u$ — $u = 3$ — is intended.", "---", "### Final Result", "The average, computed under constraints and matching the implied uniqueness condition, is:\n[\n\boxed{13}\n]", "---", "This example demonstrates how algebraic simplification, expression evaluation, and logical constraint application combine to deliver a clear, verifiable solution — perfect for educational SEO content aimed at students and math learners seeking step-by-step guidance."]