It's good that you need the understanding and not just the answer. I will start from the beginning to make sure you have all the information. If things are not clear, just post what bothers you.
8p+5q-9(8p-8q+9r) is an example of an algebraic expression. We will dissect it into smaller pieces.
8p is a term, so is 5q. An algebraic when expanded will not contain parentheses, and thus will contain nothing but terms. In the expanded form, terms are separated by + or - signs. However, the + or - signs form part of the term that follows.
Each term consists of a sign, a coefficient, and one or more variables.
The sign will always appear if the term is negative, but sometimes missing if it is positive. It can be omitted at the beginning of an expression, or at the beginning inside parentheses.
Find the two terms where the signs are omitted.
The coefficient is usually placed between the sign and the variable, and consists generally of pure numeric values. In the term 9p, 9 is the coefficient, with an omitted + sign.
In fact, the sign can be considered as part of the coefficient.
Note also that when the coefficient is "one", it will be omitted. For example, -x mean -1*x, and has to be taken into consideration when adding or subtracting coefficients.
Then comes the variable, it can be x, y, z, p, q, r, usually it is a single letter that represent a variable. A multiplication sign is implicit between the coefficient and the variable. For example, 9p means +9 * p, where * means multiplication.
Inside of a term, it is appropriate to arrange the variables in alphabetical order. For example, 4xy is equal to 4*x*y, thus equals also 4*y*x. Ordering facilitates the searching of like terms.
Like terms are terms that have the same variables raised to the same power. For example, 3xy² and -5y²x are like terms, because the second term can be rearranged to form -5xy².
Identifying like terms is critical in simplifying expressions.
For example, 4x+5y-2x can be grouped by like terms, 4x-2x+5y, since 4x and -2x are like terms.
Like terms can be combined together by doing arithmetic with the coefficients. In the above example, 4x-2x simplifies to (4-2)x=2x, or think of 4 oranges - 2 oranges = 2 oranges.
So far so good, if the expression is expanded. If the expression contains parentheses, they need to be removed according to the law of distributivity, namely a(b+c) = ab+ac.
= (-5)*4x + (-5)*(-2y)
= -20x + 10y
So 8p+5q-9(8p-8q+9r) will expand to
=8p + 5q + (-9)*(8p) + (-9)*(-8q) + (-9)*(9r)
= 8p + 5q + (-72p) + (72q) + (-81r)
= 8p + 5q - 72p + 72q - 81r
= 8p-72p + 5q+72q - 81r
If you have followed so far, can you try to finish the simplification?
If you have doubts somewhere, post so I can help you exactly where you need it.
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