Problem 84 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 6 a^{2}+18 a-9 a+5 $$

Step-by-Step Solution

Answer

The simplified expression is \(6a^2 + 9a + 5\).

Step 1: Identify Like Terms

Look at the terms in the expression: \(6a^2\), \(18a\), \(-9a\), and \(5\). The terms with \(a\) are \(18a\) and \(-9a\), which are like terms, while \(5\) is a constant term and \(6a^2\) is a term with \(a^2\).

Step 2: Combine Like Terms

Add or subtract the coefficients of the like terms. Combine \(18a\) and \(-9a\):\[ 18a - 9a = 9a \]

Step 3: Write the Simplified Expression

Reassemble the expression with the combined terms:\[ 6a^2 + 9a + 5 \]No further like terms exist, so this is the simplified expression.

Key Concepts

Like Terms

In algebra, "like terms" refers to terms that have the same variable raised to the same power. For example, in the expression given in the exercise, we have terms like \(18a\) and \(-9a\). Both of these terms are like terms because they have the variable \(a\) raised to the power of 1. Like terms are crucial because only these can be combined during simplification, which helps reduce and simplify algebraic expressions.

• If two terms have the same variable and same exponent, they are like terms.
• Like terms can be combined through addition or subtraction of their coefficients.
• In the expression \(6a^2 + 18a - 9a + 5\), the like terms are \(18a\) and \(-9a\).

Recognizing like terms is the first step in simplifying an algebraic expression, which makes the process of solving or simplifying equations easier and more efficient.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions. By recognizing and combining like terms, the expression becomes simpler and easier to understand or solve. In the expression provided, the like terms are \(18a\) and \(-9a\). These terms can be combined because they share the same variable:

• Add the coefficients of the like terms together.
• In this case, \(18 - 9 = 9\), so combining \(18a\) and \(-9a\) results in \(9a\).
• Once combined, rewrite the expression with the combined term.

After combining these terms, the original expression \(6a^2 + 18a - 9a + 5\) simplifies to \(6a^2 + 9a + 5\). This simplification makes further calculations more manageable and reduces the chance of errors.

Algebraic Terms

An algebraic term includes numbers, variables, and exponents that are combined through multiplication or division. In algebra, recognizing the role of each in a term helps in identifying which terms can be combined.

• An algebraic term consists of a numerical coefficient and a variable part raised to an exponent, such as \(6a^2\).
• Terms can be constants (like \(5\)), variables, or a product of both.
• In an expression, terms are separated by plus or minus signs.

Understanding the components of algebraic terms is fundamental for simplifying expressions. Knowing, for example, that \(6a^2\) in the expression \(6a^2 + 18a - 9a + 5\) is not a like term with \(18a\) or \(-9a\) helps prevent mistakes in simplification. Recognizing constants and similar variables is key to successfully working with algebraic expressions.