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A variable is a symbol, usually a letter like x, y, or z, that represents an unknown value(s) in mathematics. In equations, variables are used to hold the place for numbers that we don't know yet. They can change and take on different values depending on the problem or situation.

Variables are used in various branches of mathematics, including algebra, calculus, and more. They allow us to represent unknown quantities in an expression or equation, which can then be solved or manipulated. Variables help simplify complex mathematical expressions and make it easier to solve problems systematically.

Here are a few examples of variables in mathematical expressions and equations: (1) y = 2x + 3: In this linear equation, x and y are variables representing unknown values. By solving the equation for specific values of x, we can find the corresponding value of y. (2) f(x) = x^2: In this function, x is the input variable, and f(x) represents the output value, which depends on the input x. The equation describes a quadratic function, where the output is obtained by squaring the input value. (3) ax^2 + bx + c = 0: This is a general form of a quadratic equation. Here, x is the variable while a, b, and c are constants. Solving this equation will give us the possible values of x that satisfy the equation. In each example, variables help represent unknown values and allow us to manipulate and solve mathematical expressions and equations systematically.