Problem 53 - Prealgebra

Question

Use the distributive property to combine similar terms. \(-8 a-2 a\)

Step-by-Step Solution

Answer

The combined expression is \(-10a\).

Step 1: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In the expression \(-8a - 2a\), both terms are like terms because they have the same variable \(a\).

Step 2: Apply the Distributive Property

The distributive property allows us to factor out the common variable by combining like terms. Add the coefficients of the like terms: \(-8a - 2a = (-8 - 2) \cdot a\).

Step 3: Perform the Addition

Add \(-8\) and \(-2\) together: \(-8 - 2 = -10\). This gives us the combined coefficient for the like terms.

Step 4: Write the Final Expression

Substitute the combined coefficient back into the expression with the variable: \(-10a\). The simplified expression using the distributive property is \(-10a\).

Key Concepts

Like Terms

Like terms are an essential concept in algebra that help simplify expressions. They refer to terms that have the exact same variable, raised to the exact same power, even if the coefficients (the numbers in front of the variables) are different. For example, in the expression

• \(-8a - 2a\), both terms are like terms because they have the variable \(a\) raised to the first power.

Recognizing like terms is a crucial step in dealing with algebraic expressions as it allows you to simplify them effectively.

Combine Similar Terms

Combining similar terms is all about taking terms that are alike and adding or subtracting their coefficients. It makes solving algebraic expressions simpler and more efficient. Imagine you're combining like terms in an equation; this process involves

• Looking at the coefficients of the like terms: For \(-8a - 2a\), you would look at \(-8\) and \(-2\).
• Then, you simply perform the addition or subtraction: \(-8 - 2 = -10\).

By combining these like terms, you create a cleaner, more manageable form of the initial expression, turning it into \(-10a\). This step is essential in simplifying expressions, which leads us to our next concept.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as compact and straightforward as possible. This involves

• Recognizing and combining like terms through addition or subtraction of their coefficients as demonstrated.
• Using properties of mathematics such as the distributive property to rewrite expressions compactly.

In the example of

• \(-8a - 2a\), we first recognized both terms as like terms.
• Then, we combined them by performing \(-8 - 2\), resulting in a simpler \(-10a\).

Simplifying is a reliable strategy not only in solving problems but also in making equations easier to understand. It strips away unnecessary complexity, while maintaining the equivalence of the expression, preparing it for further operations if needed.