Question

In each case, simplify the given expression, if possible. \(8 p q+11 p q-9 p q\)

Short answer

Question: Simplify the expression: \(8pq + 11pq - 9pq\) Answer: The simplified expression is \(10pq\).

Step-by-step solution

Identify the like terms

In the given expression, \(8pq\), \(11pq\), and \(-9pq\) are the like terms, as they all have the same variables, p and q with the same powers.

Combine the like terms

To combine the like terms, add (or subtract) their coefficients. So, we need to calculate \(8+11-9\): \(8+11-9=10\)

Write the simplified expression

Multiply the combined coefficient (10) by the variables (pq) to get the simplified expression: \(10pq\) Therefore, the simplified expression for the given problem is \(10pq\).

Key concepts

like terms

In algebra, identifying "like terms" is a crucial skill for simplifying expressions. Like terms are terms in an algebraic expression that have identical variables raised to the same powers. For example, in the expression \(8pq + 11pq - 9pq\), each term contains the variables \(p\) and \(q\) raised to the same power (which is implicitly 1 here). This makes these terms "like terms."

Recognizing like terms is your first step. It allows you to apply operations like addition or subtraction on these terms easily. When variables and their powers match across terms, they simply need combining, which leads us to the next crucial concept.

combining coefficients

"Combining coefficients" refers to the process of adding or subtracting the numerical prefixes of like terms. Coefficients are the numbers that multiply the variables in a term. For the expression \(8pq + 11pq - 9pq\), the coefficients are 8, 11, and -9.

To simplify the expression, you need to sum these coefficients:

• First, add the positive coefficients: \(8 + 11 = 19\)
• Then, subtract the negative coefficient: \(19 - 9 = 10\)

Combining coefficients streamlines an expression and lays the groundwork for complete algebraic simplification, reducing it into a simpler, often single-term expression.

algebraic simplification

Algebraic simplification is the process of rewriting an expression in its simplest form. After identifying like terms and combining their coefficients, the next step is to express the result clearly.

In the given expression \(8pq + 11pq - 9pq\), after combining the coefficients to get 10, the variables follow in the expression. Thus, the simplified form is \(10pq\).

The goal in simplification is often not just for a cleaner look, but also to make future calculations easier. A simplified expression is more manageable and often reduces potential calculation errors in further algebraic manipulations.