Problem 65 - Elementary Algebra

Question

Simplify. $$ -14(2 x+7)+6(2 x+7) $$

Step-by-Step Solution

Answer

The simplified expression is \(-16x - 56\).

Step 1: Distribute the Constants

First, distribute the constants outside the parentheses to each term inside the parentheses. For \(-14(2x + 7)\), distribute \(-14\), and for \(6(2x + 7)\), distribute \(6\):\(-14(2x + 7) = -14 \cdot 2x + (-14) \cdot 7\)\(6(2x + 7) = 6 \cdot 2x + 6 \cdot 7\).

Step 2: Calculate the Products

Now calculate the products obtained from distribution:\(-14 \cdot 2x = -28x\)\(-14 \cdot 7 = -98\)\(6 \cdot 2x = 12x\)\(6 \cdot 7 = 42\).

Step 3: Combine Like Terms

Combine the expressions from the distribution:\(-28x - 98 + 12x + 42\).Now, combine the like terms:1. Combine the \(x\) terms: \(-28x + 12x = -16x\).2. Combine the constant terms: \(-98 + 42 = -56\).

Step 4: Write the Simplified Expression

The simplified expression after combining all like terms is:\(-16x - 56\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. It allows us to distribute a single term across terms inside parentheses. This means that if you have a term multiplied by an expression inside the parentheses, you multiply the term with each part of the expression.For example, in the expression \(-14(2x + 7)\), the \(-14\) needs to be multiplied by \(2x\) and \(7\) separately. This results in:

• \(-14 imes 2x = -28x\)
• \(-14 imes 7 = -98\)

Similarly, \(6(2x + 7)\) means multiplying the \(6\) with the \(2x\) and \(7\), which gives:

• \(6 imes 2x = 12x\)
• \(6 imes 7 = 42\)

This step is crucial as it breaks down the expression into simpler parts.

Combining Like Terms

Once you've applied the distributive property, the next step is to combine like terms. Like terms are terms in an algebraic expression that have the same variable to the same power, or are constant terms without variables. For example, \(-28x\) and \(12x\) are like terms because they both contain the variable \(x\). Combining these terms involves adding or subtracting their coefficients (the numbers in front of the variables). In this exercise:

• \(-28x + 12x = -16x\)
• The constant terms \(-98\) and \(42\) are combined to give \(-56\)

Combining like terms simplifies the expression further, making it easier to work with.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as concise and straightforward as possible. This process often involves several steps, such as using the distributive property and combining like terms, as seen in the previous sections. Let's look at the expression after distribution and combining like terms:From \(-28x - 98 + 12x + 42\), we first combine like terms to get:

• \(-16x\)
• \(-56\)

The simplified version of this expression is \(-16x - 56\). Removing any unnecessary terms or coefficients leads to minimal and more usable forms, which are especially helpful when solving equations.

Elementary Algebra

Elementary algebra involves the basic building blocks and foundational principles necessary for understanding more advanced mathematical concepts. In this particular exercise, we've touched upon some key ideas:

• The distributive property
• Combining like terms
• Simplifying expressions

These concepts are the basis for solving algebraic equations, forming the groundwork for learning advanced topics in algebra. Once you're comfortable with these principles, you can tackle more complex problems with greater ease, confidence, and efficiency. Each of these elements builds up your algebraic understanding, facilitating progress to more advanced mathematical studies.