Problem 55 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ \frac{2}{3}(12 x+15)+16 $$

Step-by-Step Solution

Answer

The simplification of the given algebraic expression is \(8x + 26\).

Step 1: Use the distributive property

Start by distributing the fraction \(\frac{2}{3}\) across the expression within the parenthesis. We get: \(\frac{2}{3} * 12x + \frac{2}{3}*15 + 16 = 8x + 10 + 16\)

Step 2: Combine the like terms

Next, combine all the similar terms (constants in this case) as follows: \(8x + (10 + 16) = 8x + 26\)

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra that helps us simplify expressions and make them easier to work with. It's a rule that allows you to multiply a single term by each term inside a set of parentheses. In our example, we start with the expression \( \frac{2}{3}(12x + 15) + 16 \). The distributive property tells us to multiply \( \frac{2}{3} \) by each term inside the parentheses: \( 12x \) and \( 15 \).

Here's how it looks step by step:

• Multiply \( \frac{2}{3} \) by \( 12x \), which gives you \( 8x \).
• Multiply \( \frac{2}{3} \) by \( 15 \), which gives you \( 10 \).

Now, after applying the distributive property, the expression becomes \( 8x + 10 + 16 \). This shows how distributing the multiplier over the terms inside the parentheses simplifies the equation.

Combining Like Terms

After using the distributive property, the next step in simplifying algebraic expressions is combining like terms. Like terms are terms in an expression that have the same variables raised to the same power. In our case, however, the only terms we need to combine are the constants because our expression now is \( 8x + 10 + 16 \).

Combining the like terms (constants) means adding them together to form a simpler expression:

• Add the constants \(10\) and \( 16 \), giving us \( 26 \).

So, the terms combine to transform the expression into \( 8x + 26 \). This operation reduces the number of terms in the expression simplifying it to its final form.

Removing Symbols of Grouping

Removing symbols of grouping involves eliminating parentheses, brackets, or any other symbols used to group elements in mathematical expressions. When simplifying expressions, the goal is to clear out these symbols while maintaining the integrity of the expression. In our example, \( \frac{2}{3}(12x + 15) + 16 \), the parentheses around \( 12x + 15 \) indicate multiplication with \( \frac{2}{3} \).

Here's the process:

• Apply the distributive property as detailed in the first section.
• Once multiplied, the parentheses are no longer needed as the operation they instructed (multiplication) has been carried out.

Effectively, using the distributive property removed the grouping symbol (parentheses), transforming the expression directly to \( 8x + 10 + 16 \) and further simplifying it by combining like terms to reach \( 8x + 26 \). This step ensures the expression is fully simplified and free of unnecessary grouping.