Algebra is a branch of mathematics that uses letters, symbols, and numbers to represent and manipulate mathematical expressions and equations. In GCSE Mathematics, algebra is an important tool for solving various mathematical problems, including word problems, equations, and inequalities.

The basic concept in algebra is using variables, usually represented by letters such as x, y, or z. Variables represent unknown values or quantities that vary in a given equation or expression.

For example, consider the equation:
3x + 5 = 20

In this equation, the variable "x" represents an unknown quantity we must solve. By using algebraic techniques, we can rearrange the equation to isolate the variable on one side of the equation:
3x + 5 - 5 = 20 - 5
3x = 15
x = 5

Here, we have solved for the value of "x" by using algebra to isolate it on one side of the equation.

Another important concept in algebra is the use of functions. A function is a relationship between two variables, usually represented by the letter "f(x)".

For example, consider the function:
f(x) = 2x + 3

In this function, the variable "x" represents the input, and the expression "2x + 3" represents the output. To evaluate the function for a specific input value, we substitute that value for "x" and simplify the expression.

For example, to evaluate the function at "x = 4", we can substitute 4 for "x" and simplify the expression:
f(4) = 2(4) + 3
f(4) = 8 + 3
f(4) = 11

Here, we have evaluated the function at "x = 4" and obtained the corresponding output value of 11.

The factorisation is another important concept in algebra, and it involves breaking down an expression into simpler terms.

For example, consider the quadratic equation:
x2 + 5x + 6 = 0