Problem 23 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 5(2 x+1)+4(3 x-2) $$

Step-by-Step Solution

Answer

The simplified expression is \(22x - 3\).

Step 1: Distribute the Numbers Inside Parentheses

First, apply the distributive property to the terms within each set of parentheses. Multiply 5 by each term in the parentheses:\[5(2x + 1) = 5 imes 2x + 5 imes 1 = 10x + 5\]Then, apply the distributive property to the second set of parentheses:\[4(3x - 2) = 4 imes 3x + 4 imes (-2) = 12x - 8\]

Step 2: Combine Like Terms

Now that we have removed the parentheses by distributing, combine the like terms to further simplify the expression. Start with combining the terms that contain \(x\):\[10x + 12x = 22x\]Next, combine the constant terms:\[5 - 8 = -3\]The expression now becomes:\[22x - 3\]

Key Concepts

Distributive Property

To understand the process of simplifying algebraic expressions, one key concept you must grasp is the distributive property. It states that for any number or variable, if you have an expression in the form of \(a(b + c)\), you can distribute \(a\) across the terms inside the parentheses. The result would be \(a \times b + a \times c\). This property helps to eliminate parentheses and set the stage for further simplification.

In the case of the exercise \(5(2x + 1) + 4(3x - 2)\), the distributive property is applied by:

• Multiplying 5 by each term in the first parentheses: \(5 \times 2x + 5 \times 1 = 10x + 5\)
• Multiplying 4 by each term in the second set: \(4 \times 3x + 4 \times (-2) = 12x - 8\)

By utilizing this concept, we can start simplifying the expression, turning a complex group into smaller, manageable parts. Once the distribution is completed, the expression is free from parentheses and ready for further steps.

Combining Like Terms

Once you have used the distributive property to remove parentheses and reorganize the expression, the next step is to combine like terms. These are terms in the expression that contain the same variable to the same power. In simple terms, like terms have the same 'ingredients.'

In our exercise, after distributing, the expression becomes \(10x + 5 + 12x - 8\). Here, the like terms involving variable \(x\) are \(10x\) and \(12x\). You can add these together:

• \(10x + 12x = 22x\)

The constant terms \(5\) and \(-8\) can also be combined:

• \(5 - 8 = -3\)

By methodically combining these terms, you streamline the expression, making it simpler and more readable. This process is crucial for reaching the most reduced form of the algebraic expression.

Simplifying Expressions

The ultimate goal in handling algebraic expressions is to simplify them to the most straightforward form possible. Simplifying makes expressions easier to work with and understand. Once you've applied the distributive property and combined like terms, you've already completed much of the simplification process.

Our expression from the exercise, after combining terms, is \(22x - 3\). This is a fully simplified expression because:

• It contains only non-parenthetical structure, meaning no more removable parentheses are present.
• All like terms have been combined, leaving each variable type and constant on its own.

At this point, further operations, equations, or problems involving the expression become less complex and more straightforward to solve. Remember, a simplified expression means one that is concise, clear, and without unnecessary complexity.