What is a variable? How do you distinguish a dependent variable from an independent one in a problem?

A variable is a symbol that represents a value, which could be a number, a vector, a matrix, or another mathematical object. In a problem, variables are often used to represent unknown quantities that can take on various values. Variables are usually represented by letters (e.g., x, y, z) or other symbols, such as Greek letters.

In a problem, the relationship between two variables determines whether one is considered dependent or independent. An independent variable is a variable whose value can be changed freely without affecting other variables. A dependent variable, on the other hand, is a variable whose value depends on one or more independent variables. This means that the value of a dependent variable will change if the value of an independent variable changes.

Let's consider an example problem to better understand the difference between dependent and independent variables. Suppose we want to study the factors affecting the growth of a plant. We collect data on plant height (in centimeters) after a certain number of days of growth. In this case, the independent variable is the number of days since the plant's growth. This variable can be freely changed without affecting any other variables. The dependent variable is the plant's height, which we measure in centimeters. The height of the plant depends on the number of days it has been growing, so it is a dependent variable in this problem. In summary, a variable is a symbol representing an unknown quantity in a problem, and the distinction between dependent and independent variables comes from their relationships with each other. Independent variables can be freely changed, while dependent variables' values depend on one or more independent variables.