Simplifying (35x - 35y + 14x - 2y) / 42 A Step-by-Step Guide

Introduction to Simplifying Algebraic Expressions

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. Algebraic expressions are mathematical phrases that combine variables (represented by letters), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division, etc.). The process of simplification aims to reduce an expression to its most basic and understandable form without changing its value. This involves combining like terms, applying the distributive property, and reducing fractions. In this comprehensive guide, we will delve into the step-by-step process of simplifying the algebraic expression ${\frac{35x - 35y + 14x - 2y}{42}}$. Mastering this skill is crucial for solving more complex equations and problems in algebra and beyond. By breaking down the expression into manageable steps, we can clearly see how each operation contributes to the final simplified form. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide a clear and thorough explanation. Let’s embark on this mathematical journey together and unlock the simplicity within complex expressions. Understanding how to simplify expressions not only helps in academics but also in various real-world applications, making it a valuable skill to possess.

Understanding the Given Expression: ${\frac{35x - 35y + 14x - 2y}{42}}$

The expression we aim to simplify is ${\frac{35x - 35y + 14x - 2y}{42}}$. This expression is a fraction where the numerator contains terms with variables ${x}$ and ${y}$, and the denominator is a constant, 42. To simplify this expression effectively, we must first identify and combine like terms in the numerator. Like terms are terms that contain the same variable raised to the same power. In our case, we have terms with ${x}$ and terms with ${y}$. The presence of both variables and a constant denominator makes this a typical algebraic simplification problem. Understanding the structure of the expression is the first step towards simplification. It allows us to strategize the order of operations and the techniques we will apply. Recognizing the components—variables, constants, and operations—is essential in tackling more complex algebraic problems in the future. The goal is to manipulate the expression while maintaining its mathematical integrity, ensuring that the simplified form is equivalent to the original. The fraction bar indicates that the entire numerator is to be divided by the denominator, which is a critical aspect to keep in mind as we proceed with the simplification steps. Thus, a clear understanding of the expression's components sets the stage for a methodical and accurate simplification process.

Step 1: Combining Like Terms in the Numerator

The first critical step in simplifying the expression ${\frac{35x - 35y + 14x - 2y}{42}}$ is to combine the like terms in the numerator. Like terms are those that have the same variable raised to the same power. In this case, we have terms with the variable ${x}$ and terms with the variable ${y}$. The ${x}$ terms are ${35x}$ and ${14x}$, and the ${y}$ terms are ${-35y}$ and ${-2y}$. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variables). So, we add ${35x}$ and ${14x}$, which gives us ${49x}$. Similarly, we combine ${-35y}$ and ${-2y}$, which results in ${-37y}$. After combining like terms, the numerator becomes ${49x - 37y}$. This step is crucial as it reduces the complexity of the expression, making it easier to proceed with further simplification. By grouping and combining like terms, we are essentially streamlining the expression, removing redundancies, and paving the way for the next steps. This process is fundamental in algebra and is used extensively in solving equations and simplifying more complex mathematical expressions. Therefore, mastering the skill of combining like terms is essential for anyone delving into algebra.

Step 2: Rewriting the Expression with Simplified Numerator

After combining the like terms in the numerator, our expression ${\frac{35x - 35y + 14x - 2y}{42}}$ transforms into ${\frac{49x - 37y}{42}}$. This step is straightforward but essential as it presents the expression in a more concise form. The simplified numerator, ${49x - 37y}$, clearly shows the combination of the ${x}$ and ${y}$ terms. Now, the expression consists of a fraction with a binomial (a two-term expression) in the numerator and a constant in the denominator. This form allows us to see the expression's structure more clearly and consider our next steps for further simplification. Rewriting the expression after each simplification step is a good practice as it helps maintain clarity and reduces the chance of errors. It also allows us to reassess the expression and determine the most efficient path forward. In this case, having a simplified numerator prepares us for the possibility of factoring or simplifying the fraction further. Thus, this step, though seemingly simple, is a crucial part of the overall simplification process.

Step 3: Checking for Common Factors and Simplifying the Fraction

Now that we have the expression ${\frac{49x - 37y}{42}}$, the next crucial step is to check for any common factors between the numerator and the denominator. This involves examining the coefficients (49 and -37) in the numerator and the constant (42) in the denominator to see if they share any divisors greater than 1. If we find a common factor, we can simplify the fraction by dividing both the numerator and the denominator by that factor. This process reduces the expression to its simplest form. In our case, the coefficients in the numerator are 49 and -37, and the denominator is 42. The factors of 49 are 1, 7, and 49. The factors of 37 are 1 and 37 (since 37 is a prime number). The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. Comparing these factors, we see that 49 and 42 share a common factor of 7. However, 37 does not share any factors with 42 other than 1. This means that we cannot directly divide the entire numerator by a common factor to simplify the fraction further. Since there isn't a common factor that can divide both terms in the numerator and the denominator, the fraction is already in its simplest form. This result is important because it tells us that we have simplified the expression as much as possible. Recognizing when an expression is in its simplest form is a key skill in algebra. It prevents unnecessary steps and ensures that we arrive at the most concise representation of the original expression.

Step 4: Final Simplified Expression

After carefully checking for common factors and determining that there are none that apply to the entire numerator and the denominator, we arrive at the final simplified expression. Our journey from the initial expression, ${\frac{35x - 35y + 14x - 2y}{42}}$, through combining like terms and checking for common factors, leads us to ${\frac{49x - 37y}{42}}$. This is the most simplified form of the given algebraic expression. There are no further simplifications possible as the terms in the numerator do not share any common factors with the denominator. The final expression clearly presents the relationship between the variables ${x}$ and ${y}$, and the constant denominator. It is crucial to recognize when an expression is in its simplest form, as this indicates that the simplification process is complete. This final step not only provides the answer but also reinforces the entire process of algebraic simplification. Understanding that each step—combining like terms, rewriting the expression, and checking for common factors—contributes to the final result is essential. The simplified expression ${\frac{49x - 37y}{42}}$ is now ready to be used in further calculations, equations, or mathematical contexts, demonstrating the power of simplification in making complex expressions more manageable.

Conclusion: Importance of Simplifying Algebraic Expressions

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics, enabling us to transform complex mathematical phrases into more manageable and understandable forms. Throughout this guide, we have meticulously simplified the expression ${\frac{35x - 35y + 14x - 2y}{42}}$, demonstrating the critical steps involved: combining like terms, rewriting the expression, and checking for common factors. The final simplified form, ${\frac{49x - 37y}{42}}$, showcases the elegance and efficiency of algebraic simplification. This process not only makes expressions easier to work with but also provides a clearer understanding of the relationships between variables and constants. The ability to simplify algebraic expressions is crucial for various mathematical applications, from solving equations to more advanced topics like calculus and linear algebra. Moreover, the skills developed through simplification extend beyond mathematics, fostering logical thinking and problem-solving abilities applicable in diverse fields. Mastering algebraic simplification equips individuals with a powerful tool for tackling mathematical challenges and enhances their overall mathematical proficiency. The steps and principles discussed in this guide serve as a solid foundation for further exploration of algebra and its applications. Thus, the importance of simplifying algebraic expressions cannot be overstated, as it forms a cornerstone of mathematical understanding and proficiency.