Independent and dependent variables Knowledge-Based Study

Study notes

1. What Are Variables in a Relationship?

   In algebra, when two quantities change together, we describe their relationship using independent and dependent variables.

   • Independent Variable: The quantity that can be changed freely or chosen first. It is the input or cause in the relationship.
   • Dependent Variable: The quantity whose value changes in response to the independent variable. It is the output or effect.

   A helpful way to remember the relationship is: The dependent variable depends on the independent variable.

   Graphing Variables

   In the standard Cartesian coordinate system, each axis is assigned a specific role:

   • The horizontal axis (the $x$-axis) represents the independent variable (input).
   • The vertical axis (the $y$-axis) represents the dependent variable (output).

   Every point on a graph is written as an ordered pair $(x, y)$, showing how a particular input value determines the corresponding output value.

   For example, in the linear equation $y = 3x + 2$, $x$ is the independent variable and $y$ is the dependent variable. Choosing a value for $x$ directly determines the value of $y$.

   Identifying Variables in Context

   Consider the following situation: A taxi service charges a flat fee of $3.00 plus$2.00 for each mile driven.

   • Distance driven (in miles): Independent variable ($x$), because the passenger chooses how far to travel.
   • Total fare (in dollars): Dependent variable ($y$), because the total cost depends on the number of miles driven.

   We can model this situation with the linear equation: $$y = 2x + 3$$

   If a passenger travels $x = 5$ miles, the cost is: $$y = 2(5) + 3 = 10 + 3 = 13$$ The total fare of $13 depends directly on the 5 miles traveled.

   Key Takeaways

   • Always ask: "Which quantity naturally affects the other?" Time, distance, or number of items purchased are often independent variables.
   • Cost, total earnings, or total distance covered over time are typically dependent variables.
   • Common Misconception: Assuming the variable written on the left is always independent. In $y = mx + b$, $y$ is isolated on the left, but it is the dependent variable because its value is calculated from $x$.