Problem 62 - Elementary Algebra

Question

For the following problems, note how many: $$ (a+5) \text { 's in } 2(a+5) ? $$

Step-by-Step Solution

Answer

Answer: There are 2 (a+5) terms present in the expression 2(a+5).

Step 1: Identify the expression

We have an algebraic expression, $$ 2(a+5) $$

Step 2: Applying distributive property

According to the distributive property in algebra, we can distribute the coefficient (2 in this case) to each term inside the parentheses: $$ 2(a+5) = 2 \times a + 2 \times 5 $$

Step 3: Simplify the expression

Now simplify the distributed terms: $$ 2 \times a + 2 \times 5 = 2a + 10 $$

Step 4: Identify how many (a+5)s are in the expression

Comparing the simplified expression to the original (a+5), we see that there are 2(a) terms and only 1(5) term. Therefore, there are 2 '(a+5)'s in the expression '2(a+5)'.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions in which a number or term is multiplied by a sum or a difference within parentheses. This property states that multiplying a sum by a number is the same as multiplying each addend by that number and then adding the products. This is mathematically expressed as:

• \( a(b+c) = ab + ac \)

For example, in the expression \(2(a+5)\), the 2 is the multiplier distributed to both \(a\) and 5. This changes the original expression inside the parentheses, \(a+5\), into two separate products: \(2 \cdot a\) and \(2 \cdot 5\).
After applying the distributive property, the expression becomes \(2a + 10\). This technique is very useful in simplifying complex algebraic expressions and is often employed to make calculations easier and more manageable.

Simplifying Expressions

Simplifying expressions is all about making the algebraic expressions look simpler while maintaining their equality. Once you've applied the distributive property, the next step is to simplify the resulting terms by performing the arithmetic operations. In our example:

• We distribute: \(2(a+5) = 2a + 10\).
• The expression is already in its simplest form as there are no like terms to combine.

Simplification involves combining like terms, which are terms that have exactly the same variable raised to the same power. In this case, there are no like terms that need combining in \(2a + 10\), as they are already reduced. Simplifying makes it easier to understand the core components of an expression, leading to a clearer and more concise result.

Algebraic Terms

Algebraic terms are the basic building blocks of expressions. They consist of coefficients, variables, and sometimes constants. Each term in an expression plays a specific role:

• The coefficient is the numerical part of a term. For example, in the term \(2a\), 2 is the coefficient.
• The variable is the letter that represents a number that can change. In both terms \(2a\) and \(10\), \(a\) is the variable term.
• The constant term has no variable attached, such as 10 in the term \(2a + 10\).

Understanding each part of a term helps in comprehending how expressions are formed and manipulated. When looking at expressions such as \(2(a+5)\), it's crucial to identify and work with each term effectively to ensure accurate mathematical calculations and simplification processes.