In math, a variable is a concept usually represented by a letter, which stands for a number, set of numbers, or range of values. For example, in the expression x + 4, the x is a variable unlike the 4, which represents a specific number.
A variable can take several different roles:
- as an unknown. For example, 3 + x = 5. This use of x asks for all members of a replacement set which make that open statement (the equation) true.
- as a generalization. Rather than give infinitely many examples of the commutative property with different numbers, we can say x + y = y + x and that takes care of them all.
- as an independent or dependent variable. In a function, one variable is the dependent variable and all the other variable(s) are independent. The dependent variable depends on the choice of the independent variables, which can be any value in the domain. For y = 2x + 1, x is independent and y is dependent. When trying to graph by creating ordered pairs, any value of x in D (the domain of this function) can be chosen, then a value for y is calculated. For z = 3x - 2y, z is dependent and x and y are independent. If x = 10 and y = 2, then z can be calculated from that. Thus the ordered triple (10, 2, 26) is a point on the function's graph.
- as a placeholder or constant, a variable simply stands for another number. In the linear equation, y = mx + b, the letters x and y are variables - they represent a range of values. The letter b represents the y-intercept (y-coordinate for point at which the line crosses the y-axis) and this is a constant value for all the values of x and y for this equation. The letter m represents the slope (or gradient) of the line represented by this equation. It is also constant for all values of x and y for a particular line. For example, in y = 2x + 3, for all the values of x and y, the slope, 2, and the y-intercept, 3, remain constant.