4. \( \frac{2}{3} v+\frac{1}{4} u \)

Did you know that expressions involving variables and coefficients have been a part of mathematics for centuries? The use of letters to represent numbers dates back to ancient civilizations, including the Egyptians and Greeks. It wasn't until the 16th century that algebra gained popularity in Europe. This expression you’ve provided is a classic example of linear combinations in algebra, where coefficients are used to represent the relationship between two variables. In real-world applications, expressions like \( \frac{2}{3} v + \frac{1}{4} u \) can represent multiple scenarios, such as calculating the total cost of items where \( v \) and \( u \) are quantities of products and \( \frac{2}{3} \) and \( \frac{1}{4} \) represent per-unit costs. This demonstrates the power of algebra in making effective decisions in budgeting and resource management.