Coefficients are basically constant multipliers attached to a variable.

Let’s say you have a term 4x. The variable is x. The term attached to x, which is the constant multiplier 4, is called the coefficient of the variable. As we noted, the variable is x so 4 is the coefficient of x in the term 4x.

Likewise, in the term -2y, the variable is y, while -2, the constant multiplier attached to y, is called the coefficient of y.

Combine them together by addition and you will have an expression 4x - 2y containing two terms 4x and -2y. The coefficient of x in this expression is 4 and the coefficient of y in the same expression is -2.

Suppose you have an expression 3x + 5x. We do not say that 3 and 5 are the coefficients of x, because the expression can be simplified to become 8x, so the coefficient of x is 8.

For the term 3xy, which is the product of 3, x and y, we say that 3 is the coefficient of xy. However, do note that this does not count as a coefficient of x, since there are other variables attached to x.

Even for the term 3x^2, we say that 3 is the coefficient of x^2, and not the coefficient of x, since the variable x is “attached to variable x”. In mathematics, even if the term does not contain other variables than x, the power of x also matters.

So in an expression such as 6x^2 - 7xy + 0.5x + 5y + 2y + 5, we first must check that it is simplified already, and this is not the case, so we get 6x^2 - 7xy + 0.5x + 7y + 5. Now we can say that the coefficient of x^2 is 6, the coefficient of xy is -7, the coefficient of x is 0.5, the coefficient of y is 7 and the constant term (the one not attached to any variable) is 5.

There are many more examples, and while this seems to be superficial, this knowledge is required in Sec 3 and Sec 4 E and A Maths because you will see the term coefficient very often.