Problem 83 - Elementary Algebra Within Reach

Question

In Exercises \(75-86\), simplify the expression. $$ -4(2-5 x)+3(x+6) $$

Step-by-Step Solution

Answer

The simplified expression of the problem is \(23x + 10\).

Step 1: Distribute the numbers to eliminate parentheses

To start with, -4 multiplied with each term inside the first parentheses: \( -4 \times 2 \) and \( -4 \times -5x \). Similarly, 3 is multiplied with each term inside the second parentheses: \( 3 \times x \) and \( 3 \times 6 \). Thus, the expression becomes: \[ -4 \times 2 - 4 \times -5x + 3 \times x + 3 \times 6 \]

Step 2: Perform the multiplication

Next, we perform the multiplication operations: \[ -8 + 20x + 3x + 18 \]

Step 3: Combine like terms

The final step is to combine the like terms. Here, 20x and 3x are like terms, and -8 and 18 are like terms. Thus, the final expression becomes: \[ 23x + 10 \]

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The beauty of algebraic expressions lies in their ability to represent relationships between quantities that may vary. Think of it like a recipe: just as a recipe outlines the ingredients to create a dish, an algebraic expression combines numbers and variables to depict a mathematical situation. For instance, in our exercise, \( -4(2-5x)+3(x+6) \) is an algebraic expression that tells us how quantities related to 'x' are mixed together using multiplication, subtraction, and addition.

Learning to work with these expressions is essential as they form the foundation of algebra and appear in almost every aspect of more advanced mathematics. Simplifying an algebraic expression, which involves reducing it to its simplest form, makes it easier to understand and work with. This is analogous to simplifying a complex recipe into basic steps, allowing us to see the core ingredients and their quantities more clearly.

Distributive Property

The distributive property is a cornerstone of algebra, and it allows us to distribute a single term over terms inside a parenthesis. The formula for the distributive property is: \( a(b + c) = ab + ac \). It's similar to handing out flyers; if we have one flyer (the 'a') and we need to give it to two people (the 'b' and 'c'), each person gets a flyer. In our exercise, we applied the distributive property when we multiplied -4 by both 2 and -5x, as well as 3 by x and 6. Here's a breakdown:

• \( -4 \times (2 - 5x) \) becomes \( -4 \times 2 - 4 \times -5x \)
• \( 3 \times (x + 6) \) becomes \( 3 \times x + 3 \times 6 \)

Understanding and correctly applying the distributive property is crucial, as it enables us to modify expressions so that we can simplify them further by combining like terms.

Combining Like Terms

When simplifying algebraic expressions, one key step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, 2x and 5x can be combined because they both have the single variable 'x' raised to the first power. However, 2x and 2x^2 cannot be combined as their exponents differ.

In the context of our example, after distributing, we had the terms 20x and 3x, which are like terms. We also had constants -8 and +18, which are considered like terms because they do not contain any variables. By combining them, we simplify the expression to a more manageable form:

• The like terms 20x and 3x combine to become 23x
• The constant terms -8 and +18 combine to 10

Thus, the simplified expression is \( 23x + 10 \). Recognizing and combining like terms effectively is a vital skill in algebra, as it simplifies expressions and equations, making them easier to interpret and solve.

Elementary Algebra

The study of elementary algebra is the stepping stone to more complex areas of mathematics, but at its heart, it's about understanding how to manipulate algebraic expressions and equations. This includes simplifying expressions, solving for variables, and working with different forms of algebraic equations. In the exercise, we've been using principles of elementary algebra: distributing numbers over parentheses, multiplying terms, and combining like terms.

Why is it important?

Mastering elementary algebra equips you with a problem-solving toolset that applies to countless situations beyond the classroom, from calculating finances to understanding scientific formulas. The exercise we've gone through exemplifies how algebra is not just about finding a solution, but also about appreciating the logical process that leads to that solution. By taking the time to understand this foundational level of algebra, students build a strong base for all future mathematical learning and application.