Solving Algebraic Equations A Step By Step Guide With Karissa's Solution

In the realm of mathematics, solving equations is a fundamental skill that requires a blend of algebraic manipulation and logical reasoning. In this article, we will delve into the intricacies of Karissa's approach to solving the equation $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$. By meticulously examining each step of her work, we aim to illuminate the underlying principles and techniques involved in equation solving. This exploration will not only provide a clear understanding of Karissa's method but also serve as a valuable guide for anyone seeking to enhance their equation-solving prowess.

Karissa embarks on a journey to solve the equation $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$. This equation, at first glance, might appear complex due to the presence of fractions and parentheses. However, with a systematic approach, it can be simplified and solved effectively. Karissa's work, which we will dissect step-by-step, showcases the power of algebraic manipulation in unraveling such equations. From distributing coefficients to combining like terms, each operation plays a crucial role in isolating the variable and ultimately finding the solution. We'll break down each step, explaining the mathematical rationale and highlighting key concepts such as the distributive property, the combination of like terms, and the importance of maintaining balance on both sides of the equation. This deep dive will provide a comprehensive understanding of how to tackle similar equations and instill confidence in your own problem-solving abilities. Furthermore, we will connect these techniques to broader mathematical principles, demonstrating how equation solving is a cornerstone of algebra and its applications in various fields. By the end of this analysis, you will not only understand the specific steps Karissa took but also the general strategies for approaching and conquering algebraic equations.

Step 1 Distributing the $\frac{1}{2}$

Karissa's initial step involves distributing the $\frac{1}{2}$ across the terms within the parentheses $(x-14)$. This is a crucial application of the distributive property, a cornerstone of algebraic manipulation. The distributive property states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In this case, $a$ is $\frac{1}{2}$, $b$ is $x$, and $c$ is $-14$. Therefore, $\frac{1}{2}(x-14)$ becomes $\frac{1}{2}x - \frac{1}{2}(14)$, which simplifies to $\frac{1}{2}x - 7$. This step is pivotal because it eliminates the parentheses, making the equation easier to work with. It transforms the expression from a product of a number and a binomial into a sum of individual terms, each of which can be manipulated independently. By correctly applying the distributive property, Karissa lays the groundwork for subsequent simplifications. Failing to distribute correctly at this stage would propagate errors throughout the rest of the solution, highlighting the importance of mastering this fundamental algebraic technique. This first step not only simplifies the equation but also exemplifies the strategic approach required in problem-solving: breaking down complex expressions into manageable parts. The distributive property is not just a mechanical rule; it's a tool that allows us to reshape equations and reveal underlying structures. This transformation is a classic example of how mathematical operations can expose hidden relationships and pave the way for further simplification.

This transformation is key to simplifying the equation and making it more manageable. The initial equation, $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$, contains parentheses that obscure the relationships between terms. By applying the distributive property, Karissa removes these parentheses, revealing the individual terms and their coefficients. This is analogous to clearing away obstacles to get a clearer view of the landscape. Once the parentheses are gone, the equation becomes a collection of terms that can be combined and manipulated more easily. This step demonstrates the power of algebraic tools to transform complex expressions into simpler, more transparent forms. The distributive property is not just a rule to be memorized; it's a fundamental principle that underlies much of algebraic manipulation. Its correct application is essential for solving equations accurately and efficiently. Furthermore, the act of distributing the $\frac{1}{2}$ across the parentheses sets the stage for the next steps in the solution, where like terms will be combined and the variable $x$ will be isolated. This sequential approach, where each step builds upon the previous one, is characteristic of mathematical problem-solving. Karissa's initial move showcases her understanding of the strategic importance of the distributive property in simplifying equations and paving the way for a solution.

Step 2 Combining Like Terms

Following the distribution, Karissa combines the constant terms on the left side of the equation. She has $-7$ and $+11$, which combine to give $+4$. This step, $\frac{1}{2} x - 7 + 11 = \frac{1}{2} x + 4$, exemplifies the concept of combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, $-7$ and $+11$ are both constant terms, meaning they have no variable component. Combining them involves simply adding their coefficients, which are the numerical parts of the terms. This operation streamlines the equation, reducing the number of terms and making it easier to manage. Combining like terms is a fundamental simplification technique in algebra, applicable across a wide range of equations and expressions. It's a way of tidying up the equation, making it more readable and revealing its underlying structure. By combining the constants, Karissa prepares the left side of the equation for further manipulation and moves closer to isolating the variable $x$. This step underscores the importance of recognizing and combining like terms as a core skill in algebraic simplification. The ability to efficiently combine like terms is crucial for solving equations accurately and efficiently. It prevents unnecessary clutter and allows the solver to focus on the essential relationships between the variables and constants in the equation. Karissa's execution of this step demonstrates her understanding of this fundamental principle and its application in simplifying algebraic expressions.

This simplification is not merely cosmetic; it significantly reduces the complexity of the equation and makes it easier to work with. Before combining the constants, the left side of the equation contained three separate terms: $\frac{1}{2}x$, $-7$, and $+11$. After combining $-7$ and $+11$ into $+4$, the left side now contains only two terms: $\frac{1}{2}x$ and $+4$. This reduction in the number of terms streamlines the equation and makes it easier to visualize the relationships between the variable $x$ and the constant terms. Combining like terms is a strategic step that simplifies the structure of the equation. It allows the solver to focus on the essential elements and to make informed decisions about the next steps in the solution. In this case, combining the constants sets the stage for further simplification by allowing Karissa to isolate the variable $x$ more effectively. The resulting equation, $\frac{1}{2}x + 4$, is more concise and transparent than the previous expression, making it easier to compare with the terms on the right side of the equation. This comparison will be crucial in the subsequent steps, as Karissa continues to manipulate the equation and move towards the solution. By demonstrating her ability to combine like terms, Karissa showcases a fundamental algebraic skill that is essential for solving equations accurately and efficiently.

Step 3 Simplifying the Right Side

On the right side of the equation, Karissa simplifies the expression $\frac{1}{2} x-(x-4)$. This requires distributing the negative sign (which is equivalent to multiplying by -1) across the terms within the parentheses. This yields $\frac{1}{2}x - x + 4$. Now, she combines the 'x' terms, $\frac{1}{2}x$ and $-x$. To do this, it's helpful to think of $-x$ as $-\frac{2}{2}x$. Combining these gives $\frac{1}{2}x - \frac{2}{2}x = -\frac{1}{2}x$. So, the right side of the equation simplifies to $-\frac{1}{2}x + 4$. This step is a demonstration of several important algebraic techniques. First, it showcases the distribution of a negative sign, which is crucial for correctly handling expressions with parentheses and subtraction. Second, it highlights the importance of finding a common denominator when combining fractions. By expressing $-x$ as $-\frac{2}{2}x$, Karissa makes it possible to add the 'x' terms together. Third, it underscores the need to pay close attention to signs when performing algebraic operations. A mistake in sign manipulation can lead to an incorrect solution. Karissa's careful execution of this step demonstrates her proficiency in these fundamental algebraic skills. The ability to simplify expressions involving parentheses, fractions, and negative signs is essential for success in algebra. This step not only simplifies the right side of the equation but also prepares it for comparison with the simplified left side, paving the way for further steps in solving the equation. By skillfully simplifying the right side, Karissa moves closer to isolating the variable and finding the solution.

The ability to correctly manipulate expressions with negative signs and fractions is crucial for solving algebraic equations effectively. The expression $\frac{1}{2} x-(x-4)$ presents a common challenge for students learning algebra: how to handle the subtraction of a binomial. Karissa's approach, which involves distributing the negative sign and then combining like terms, is a standard technique for addressing this type of expression. The key insight is that subtracting a quantity is the same as adding its negative. Therefore, $\frac{1}{2} x-(x-4)$ can be rewritten as $\frac{1}{2} x + (-1)(x-4)$. Distributing the -1 across the parentheses then gives $\frac{1}{2}x - x + 4$. The next challenge is to combine the 'x' terms, which requires recognizing that $-x$ is equivalent to $-\frac{2}{2}x$. This step demonstrates the importance of understanding fractions and their relationship to whole numbers. By expressing $-x$ as $-\frac{2}{2}x$, Karissa creates a common denominator, allowing her to combine the 'x' terms. The result, $-\frac{1}{2}x$, is a crucial component of the simplified right side of the equation. Karissa's mastery of these techniques highlights the interconnectedness of different algebraic concepts. Distributing negative signs, working with fractions, and combining like terms are all essential skills that work together to simplify expressions and solve equations. This step not only simplifies the right side of the equation but also provides a valuable illustration of how these skills can be applied in practice.

Step 4 The Simplified Equation

After simplifying both sides, Karissa arrives at the equation $\frac{1}{2} x+4=-\frac{1}{2}x+4$. This equation is the result of all the previous manipulations, and it represents a significant milestone in the solution process. It's a much cleaner and more concise form of the original equation, making it easier to see the relationship between the terms and the variable. This simplified equation highlights the power of algebraic manipulation. By applying the distributive property, combining like terms, and handling negative signs and fractions, Karissa has transformed a complex equation into a more manageable one. This simplification is not just about aesthetics; it's about making the underlying mathematical structure more transparent. The simplified equation reveals that the 'x' terms appear on both sides of the equation, and the constant terms are the same on both sides. This observation provides a crucial clue about the nature of the solution. The simplified equation serves as a bridge to the final steps in the solution process. It allows Karissa to focus on isolating the variable 'x' and determining its value. The next steps will likely involve moving terms around the equation to group the 'x' terms together and the constant terms together. The simplified equation provides a solid foundation for these steps, making them more straightforward and less prone to error. By arriving at this simplified form, Karissa demonstrates her mastery of fundamental algebraic techniques and her ability to transform complex equations into simpler, more solvable forms. This skill is essential for success in algebra and in any field that relies on mathematical problem-solving.

This step underscores the importance of simplification as a strategy for solving complex problems. The original equation, $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$, is relatively complex, with parentheses, fractions, and multiple terms. The simplified equation, $\frac{1}{2} x+4=-\frac{1}{2}x+4$, is much simpler, with fewer terms and no parentheses. This simplification makes the equation easier to understand and to solve. The process of simplification involves applying a series of algebraic techniques to transform an expression or equation into a more manageable form. These techniques include distributing, combining like terms, and handling negative signs and fractions. Each of these techniques is a tool in the algebraic toolbox, and the skilled problem-solver knows how to use them effectively. Simplification is not just a matter of making the equation look nicer; it's about revealing the underlying mathematical structure. By simplifying an equation, we can often gain insights into the relationships between the variables and constants, and we can make it easier to isolate the variable and find the solution. The simplified equation, $\frac{1}{2} x+4=-\frac{1}{2}x+4$, provides a clear example of this principle. The equation's structure makes it easier to see the next steps required to solve for 'x'. Karissa's ability to simplify the equation effectively demonstrates her understanding of these algebraic principles and her skill in applying them to solve problems.

Conclusion

Karissa's journey through solving the equation $\frac{1}{2}(x-14)+11=\frac{1}{2} x-(x-4)$ showcases the power and elegance of algebraic manipulation. From the initial distribution to the final simplified equation, each step demonstrates a mastery of fundamental techniques. This step-by-step breakdown not only provides a clear understanding of Karissa's method but also serves as a valuable resource for anyone looking to improve their equation-solving skills. The distributive property, combining like terms, and careful handling of negative signs are all essential tools in the algebraist's toolkit. By diligently applying these techniques, complex equations can be transformed into simpler, more manageable forms. This process of simplification is not just about finding the solution; it's about gaining a deeper understanding of the relationships between variables and constants. Karissa's work exemplifies the importance of precision, attention to detail, and a systematic approach in mathematical problem-solving. This exploration of Karissa's solution is a testament to the beauty and power of algebra as a problem-solving tool.

The ability to solve equations is not just a mathematical skill; it's a critical thinking skill that has applications in many areas of life. Equations are used to model real-world phenomena, to make predictions, and to solve problems in fields ranging from physics and engineering to economics and finance. The process of solving an equation involves breaking down a complex problem into smaller, more manageable steps, identifying the key relationships between variables and constants, and applying logical reasoning to isolate the unknown. These skills are valuable not only in mathematics but also in any situation that requires problem-solving and decision-making. Karissa's solution provides a clear example of how these skills can be applied in practice. Her step-by-step approach, her attention to detail, and her understanding of algebraic principles all contribute to her success in solving the equation. By studying Karissa's work, we can gain insights into the problem-solving process and develop our own skills in this area. The ability to solve equations is a powerful tool, and mastering this skill can open doors to new opportunities and challenges in mathematics and beyond. Karissa's journey through this equation serves as an inspiration to continue learning and developing our mathematical abilities.