Problem 77 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$2 a+5-9 a$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression \(2a + 5 - 9a\) is \(-7a + 5\).

Step 1: Identify like terms

In the expression \(2a + 5 - 9a\), the like terms are \(2a\) and \(-9a\) because they both contain the variable 'a'.

Step 2: Combine like terms

To combine these terms, simply add the coefficients together. The coefficient of \(2a\) is 2 and the coefficient of \(-9a\) is -9. So when they are added together, the result is \(-7a\).

Step 3: Write down the final simplified expression

The final simplified expression is \(-7a + 5\). This expression cannot be simplified further because it only contains unlike terms now.

Key Concepts

Combining Like Terms

Understanding how to combine like terms is crucial when you're simplifying algebraic expressions. Like terms are terms that have exactly the same variable parts, which means they have the same letter or letters, and these letters are raised to the same power. For instance, in the given exercise, the like terms are \(2a\) and \( -9a\), both containing the variable 'a'.

When combining them, we focus only on their coefficients. The coefficient is the numerical part of a term that is multiplied by the variable. In this example, the respective coefficients are 2 and -9. By adding these numbers together, we are essentially adding or subtracting the quantities they represent, while the variable part remains unchanged. The result \( -7a\) combines the like terms into a single term, which simplifies the expression.

Practice Tip

When practicing, always look for terms that can be combined, and don't forget to consider the signs in front of each term as positive or negative values when combining.

Algebraic Coefficients

The term algebraic coefficients refers to the numbers that multiply the variables in an algebraic expression. It's very important to understand coefficients because they tell us how much of a given variable we have. In the expression from our exercise, the coefficients are the numbers 2 and -9. These numbers directly influence the value of the expression and determine how the like terms can be combined.

To accurately combine like terms, we have to add or subtract the coefficients, which essentially means we're adding or subtracting the quantities of the variables they represent. Always make sure you pay attention to the sign of the coefficient, as it indicates whether the term adds to or subtracts from the total value of the expression.

Common Mistake Alert

A common mistake is to ignore the sign of the coefficient, which can lead to an incorrect solution. Always treat the sign as part of the coefficient when combining terms.

Simplified Expression

A simplified expression means that we've reduced the expression to its most basic form, where no further combining of like terms is possible. In the context of the given exercise, after combining like terms, we obtained \( -7a + 5\). No further simplification is possible because \( -7a\) and \(5\) are not like terms; they are unrelated since one is a variable term and the other is a constant term.

In essence, simplifying an algebraic expression is like tidying up a room—everything should be in its right place and nothing unnecessary should be left over. The aim is to make the expression as straightforward and easy to work with as possible.

Thinking Ahead

After simplifying, think about whether each term serves a purpose or if the expression can be further reduced. This habit of