Independent Variable Vs. Dependent Variable

One of the most important concepts in mathematics, which you must grasp, is the idea of a 'Variable'. In your first mathematics courses, the focus will be on learning basic arithmetic and its applicability in everyday life. To set up an equation to solve word problems, you need to understand what dependent and independent variables are. The comparison presented in the following lines, will help you understand how these two concepts build an equation.

In a mathematical equation, a variable is any quantity whose value may change with time, or any other parameter. An equation is generally a relation between variables. In mathematics and statistics, these variables are generally identified with alphabets. Consider the equation:

z = x + y

Here z, x, and y are variables that satisfy the above equation. They can take various values, as long as the above relation is maintained. When solving a word problem, you need to set up an equation, where the unknown is generally denoted as a variable.

Let us now see how dependent and independent variables differ from each other. The concept can only be understood in the context of an equation as it relates different variables together. Let me define the two types, followed by the presentation of examples.

In the context of a mathematical equation, an independent variable is the quantity on which, rest of the variable values depend. The change in a system, which is being modeled by a mathematical equation, manifests itself through the independent variable, which in turn, affects the changes that happen in the rest of the system.

It is the 'prime mover' which impacts the overall behavior of the system. For example, consider the second of Newton's laws of motion taught in physics, which is represented by the equation:

F = ma

where 'm' is mass, 'a' is acceleration and 'F' is the force acting on an object. Here force is the independent variable as m is constant and the acceleration created in the object, is directly proportional to the force applied. More the force, more is the acceleration, which explains why it's the independent variable.

By now you must have already figured out what a dependent variable is. The variable whose value is decided by the independent variable, is the dependent variable. In the above example, the value of acceleration is decided by the force applied to the object, which makes acceleration to be a dependent variable.

Now that we have defined independent and dependent variables, here I present some real life examples, which will help you understand the difference between the two, in a better way.

Consider the weather thermometer, whose temperature value rises and falls according to the heat pervading its surroundings. If you analyze this system, heat is the independent variable which affects the dependent variable of temperature.

As another example, consider the profit made by the manufacturing unit, which is dependent on the sales volumes of the company. Here sales volume is the independent variable and profit is the dependent variable.

Any system you consider, has its share of independent and dependent variables if you closely scrutinize it. Solving a problem is identifying the equation that accurately describes the variation of dependent variables according to the independent variables. When you understand that equation and the nature of dependence between variables, you can safely say that you understand the system.

Setting up an equation is establishing a connection between the dependent and independent variables that govern the system. The system under consideration may be anything from the weather to a business process.