Simplify $-6(2x-9)+(7-6x)$: A Step-by-Step Solution
In the realm of mathematics, algebraic expressions often present themselves as intricate puzzles, demanding a meticulous approach to unravel their hidden solutions. The equation is a prime example, beckoning us to embark on a step-by-step journey of simplification and solution. This article serves as a comprehensive guide, dissecting the equation, elucidating the underlying principles, and arriving at the correct answer with clarity and precision. Let's dive into the world of algebraic manipulation and conquer this mathematical challenge.
The Art of Distribution: Unveiling the Components
The cornerstone of simplifying algebraic expressions lies in the art of distribution. This fundamental principle allows us to dismantle complex expressions into manageable components, paving the way for seamless simplification. In our equation, , the distribution property takes center stage, guiding us through the initial steps of our solution.
Our focus turns to the term , where the coefficient -6 acts as a multiplier for the entire expression within the parentheses. To distribute effectively, we meticulously multiply -6 by each term inside the parentheses, adhering to the rules of signed number multiplication. This process unfolds as follows:
- Multiplying -6 by 2x: The product of -6 and 2x yields -12x. Remember, the product of a negative number and a positive number is always negative.
- Multiplying -6 by -9: The product of -6 and -9 results in +54. Here, the product of two negative numbers is positive.
Thus, the distribution of -6 across the expression (2x-9) transforms it into -12x + 54. This transformation marks a significant stride in our simplification journey, as we've effectively eliminated the parentheses and revealed the individual components of this term.
Now, let's shift our attention to the second part of the equation, (7-6x). This expression, while seemingly simple, holds its own significance in the overall equation. To maintain clarity and consistency, we can consider this expression as being multiplied by an implied coefficient of 1. This subtle understanding allows us to apply the distribution property, even when it might not appear immediately necessary.
Distributing 1 across (7-6x) is a straightforward process, as multiplying any term by 1 leaves it unchanged. Therefore, 1 * 7 = 7 and 1 * -6x = -6x. This step might seem trivial, but it underscores the importance of recognizing the underlying mathematical principles at play.
With the distribution phase complete, our equation now takes a new form: -12x + 54 + 7 - 6x = . We've successfully broken down the complex expressions into their constituent terms, setting the stage for the next crucial step: combining like terms.
Combining Like Terms: Unifying the Elements
In the realm of algebraic simplification, the concept of "like terms" emerges as a powerful tool, enabling us to consolidate and streamline expressions. Like terms are those that share the same variable raised to the same power. In our equation, -12x + 54 + 7 - 6x =, the like terms stand out as -12x and -6x, both featuring the variable 'x' raised to the power of 1. Additionally, the constants 54 and 7 also qualify as like terms, as they represent numerical values without any variable component.
To combine like terms, we simply add or subtract their coefficients while preserving the variable and its exponent. Let's begin by focusing on the 'x' terms:
- Combining -12x and -6x: The sum of -12x and -6x is -18x. This is achieved by adding the coefficients -12 and -6, which results in -18, and then appending the variable 'x'.
Next, we turn our attention to the constant terms:
- Combining 54 and 7: The sum of 54 and 7 is 61. This is a straightforward arithmetic operation.
By meticulously combining like terms, we've transformed our equation into a more concise and manageable form: -18x + 61 = . This simplified expression represents the essence of the original equation, capturing its mathematical core in a succinct manner.
The Final Solution: Embracing the Answer
Having navigated the intricate steps of distribution and combining like terms, we arrive at the final solution of our equation: -18x + 61. This expression represents the simplified form of the original equation, encapsulating its mathematical truth in a clear and concise manner.
Comparing our solution with the options provided, we find that it aligns perfectly with option F. Therefore, the correct answer to the equation is indeed -18x + 61. This triumphant conclusion marks the culmination of our algebraic journey, a testament to the power of methodical simplification and the beauty of mathematical precision.
In conclusion, the equation serves as a compelling illustration of the principles of algebraic manipulation. By diligently applying the distributive property and meticulously combining like terms, we've successfully unraveled the equation, arriving at the definitive solution of -18x + 61. This process not only reinforces our understanding of algebraic techniques but also underscores the elegance and logic inherent in the world of mathematics.
Mastering Algebraic Expressions: A Guide to Success
Algebraic expressions, with their intricate combinations of variables, coefficients, and operations, often pose a challenge to students and math enthusiasts alike. However, with a systematic approach and a solid grasp of fundamental principles, mastering algebraic expressions becomes an attainable goal. This section delves into essential strategies and techniques that empower you to confidently tackle any algebraic expression that comes your way.
1. The Foundation: Understanding the Basics
Before embarking on complex manipulations, it's crucial to establish a strong foundation in the basic building blocks of algebra. This includes:
- Variables: Symbols (usually letters) that represent unknown quantities.
- Coefficients: Numerical values that multiply variables.
- Constants: Numerical values that stand alone, without any variable component.
- Operations: Mathematical processes such as addition, subtraction, multiplication, and division.
A firm understanding of these elements forms the bedrock upon which more advanced algebraic skills are built.
2. The Distributive Property: Unveiling Hidden Terms
The distributive property is a cornerstone of algebraic simplification, allowing us to expand expressions involving parentheses. This property dictates that a term multiplied by an expression within parentheses is distributed to each term inside the parentheses. For example, a(b + c) = ab + ac. Mastering the distributive property is essential for simplifying a wide range of algebraic expressions.
3. Combining Like Terms: Streamlining Expressions
Like terms, as we've discussed, are terms that share the same variable raised to the same power. Combining like terms involves adding or subtracting their coefficients while preserving the variable and its exponent. This process streamlines expressions, making them easier to understand and manipulate. For instance, 3x + 5x - 2x simplifies to 6x.
4. Order of Operations: Maintaining Mathematical Harmony
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures consistency in mathematical calculations. Adhering to this order is crucial when simplifying algebraic expressions involving multiple operations. For example, in the expression 2 + 3 * 4, multiplication takes precedence over addition, resulting in 2 + 12 = 14.
5. Recognizing Patterns: Unlocking Shortcuts
As you delve deeper into algebra, you'll begin to recognize recurring patterns and special cases. These patterns often provide shortcuts and efficient methods for simplification. Some notable patterns include:
- Difference of Squares: a² - b² = (a + b)(a - b)
- Perfect Square Trinomials: (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²
Identifying and utilizing these patterns can significantly expedite the simplification process.
6. Practice Makes Perfect: Honing Your Skills
The key to mastering algebraic expressions lies in consistent practice. Work through a variety of examples, gradually increasing the complexity. This hands-on experience will solidify your understanding of the concepts and techniques involved. Don't shy away from challenging problems; they provide valuable opportunities for growth and skill development.
7. Seek Guidance: Embracing Collaboration
When faced with challenging expressions, don't hesitate to seek guidance from teachers, tutors, or fellow students. Collaboration can provide fresh perspectives and insights, helping you overcome hurdles and deepen your understanding. Online resources, such as tutorials and forums, can also offer valuable support.
8. Stay Organized: A Key to Success
Maintaining organization is crucial when simplifying algebraic expressions, especially those involving multiple steps. Write neatly, align like terms, and clearly label each step. This methodical approach minimizes errors and enhances clarity. A well-organized solution is not only easier to follow but also reduces the likelihood of mistakes.
9. Check Your Work: Ensuring Accuracy
After simplifying an algebraic expression, it's always prudent to check your work. Substitute numerical values for the variables in both the original expression and the simplified expression. If the results match, you can be confident in your solution. This step provides a valuable safeguard against errors.
10. Embrace the Challenge: Cultivating a Growth Mindset
Mastering algebraic expressions can be challenging, but it's also incredibly rewarding. Embrace the challenge as an opportunity for growth and intellectual development. Cultivate a growth mindset, believing that your abilities can be developed through dedication and hard work. With persistence and a positive attitude, you can conquer any algebraic obstacle.
By adhering to these strategies and techniques, you can transform algebraic expressions from daunting challenges into manageable puzzles. The journey to mastery requires dedication and practice, but the rewards are well worth the effort. So, embrace the world of algebra, hone your skills, and unlock the power of mathematical expression.
Real-World Applications of Algebraic Expressions: Beyond the Classroom
Algebraic expressions, often perceived as abstract mathematical concepts confined to textbooks and classrooms, possess a remarkable versatility that extends far beyond academic settings. These expressions serve as powerful tools for modeling, analyzing, and solving real-world problems across diverse fields. This section unveils the practical applications of algebraic expressions, showcasing their relevance in everyday life and various professional domains.
1. Financial Planning: Budgeting and Investments
Algebraic expressions play a crucial role in financial planning, enabling individuals and businesses to manage their finances effectively. Budgeting, a cornerstone of financial stability, often involves creating algebraic expressions to represent income, expenses, and savings. For instance, an expression like Income - Expenses = Savings helps determine the amount available for future goals. Similarly, investment calculations rely heavily on algebraic expressions to model growth, interest rates, and returns. Understanding these expressions empowers individuals to make informed financial decisions.
2. Engineering and Physics: Modeling Physical Phenomena
In the realm of engineering and physics, algebraic expressions serve as the language for describing and predicting physical phenomena. Engineers utilize algebraic equations to design structures, analyze circuits, and optimize processes. Physicists employ algebraic expressions to model motion, forces, energy, and other fundamental concepts. For example, the equation for projectile motion, which involves algebraic terms for initial velocity, angle, and gravity, allows scientists to predict the trajectory of a projectile. The ability to manipulate and interpret algebraic expressions is paramount for professionals in these fields.
3. Computer Science: Algorithm Development and Programming
Computer science, the engine of the digital age, relies heavily on algebraic expressions for algorithm development and programming. Algorithms, the step-by-step instructions that guide computers, often involve algebraic operations and conditional statements. Programming languages, the tools used to create software, utilize algebraic expressions to define variables, perform calculations, and control program flow. A programmer's proficiency in algebraic manipulation directly impacts their ability to write efficient and effective code.
4. Economics: Supply and Demand Analysis
Economics, the study of resource allocation, employs algebraic expressions to model supply and demand relationships. Supply curves, which represent the quantity of a product or service that producers are willing to offer at various prices, are often expressed algebraically. Similarly, demand curves, which depict the quantity of a product or service that consumers are willing to purchase at different prices, can be represented algebraically. Analyzing these expressions helps economists understand market equilibrium, predict price fluctuations, and inform policy decisions.
5. Business Management: Cost Analysis and Profit Maximization
In the world of business management, algebraic expressions serve as essential tools for cost analysis and profit maximization. Businesses utilize cost functions, which are algebraic expressions that represent the total cost of production, to understand the relationship between output and expenses. Profit functions, derived from algebraic expressions for revenue and cost, help businesses determine the optimal production level to maximize profits. These analyses guide strategic decision-making, ensuring business sustainability and growth.
6. Data Analysis: Statistical Modeling and Interpretation
Data analysis, the process of extracting insights from raw data, relies heavily on algebraic expressions for statistical modeling and interpretation. Statistical models, which describe relationships between variables, often involve algebraic equations. For instance, linear regression, a widely used statistical technique, utilizes an algebraic equation to model the relationship between a dependent variable and one or more independent variables. The ability to interpret these algebraic models is crucial for data-driven decision-making across various domains.
7. Everyday Life: Problem-Solving and Decision-Making
Beyond professional applications, algebraic expressions permeate our everyday lives, aiding in problem-solving and decision-making. From calculating discounts at the store to determining the optimal route for a journey, algebraic thinking underlies many of our daily choices. Even simple tasks, such as dividing a pizza fairly among friends, involve algebraic concepts. The ability to translate real-world scenarios into algebraic expressions empowers us to navigate everyday challenges with greater confidence and efficiency.
In conclusion, algebraic expressions transcend the boundaries of mathematics classrooms, emerging as indispensable tools in a multitude of fields. Their ability to model, analyze, and solve real-world problems makes them essential for professionals and individuals alike. From financial planning to engineering design, from computer programming to everyday decision-making, algebraic expressions empower us to understand and shape the world around us. Embracing the power of algebra unlocks a world of possibilities, enabling us to approach challenges with clarity, precision, and ingenuity.
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