Solving The Equation (635)/(72) = -(5)/(2) * (-(11)/(4) + X)

Decoding the Mathematical Equation: \frac{635}{72}=-\frac{5}{2}\left(-\frac{11}{4}+x\right)

In this comprehensive exploration, we embark on a journey to decipher the intricacies of the equation ${\frac{635}{72}=-\frac{5}{2}\left(-\frac{11}{4}+x\right)}$. Our mission is to unravel the value of the unknown variable, x, by meticulously dissecting the equation and applying fundamental algebraic principles. This endeavor will not only illuminate the solution but also deepen our comprehension of equation-solving techniques, which are paramount in various mathematical and scientific domains. The ability to manipulate equations and isolate variables is a cornerstone of mathematical proficiency, enabling us to model real-world phenomena and solve intricate problems with precision and confidence. This article serves as a guiding light, illuminating each step of the solution process and fostering a robust understanding of the underlying mathematical concepts. By engaging with this detailed exploration, readers will not only gain the ability to solve this specific equation but also cultivate a broader aptitude for tackling algebraic challenges. The journey through this equation is more than just finding a numerical answer; it's an expedition into the heart of mathematical reasoning, where logic and precision converge to unlock solutions. Let's embark on this exciting voyage of mathematical discovery together, where each step taken brings us closer to unraveling the value of x and solidifying our algebraic prowess.

Step-by-Step Solution to \frac{635}{72}=-\frac{5}{2}\left(-\frac{11}{4}+x\right)

To effectively solve the equation ${\frac{635}{72}=-\frac{5}{2}\left(-\frac{11}{4}+x\right)}$, we will proceed systematically, employing a series of algebraic manipulations to isolate the variable x. Each step will be meticulously explained to ensure clarity and understanding. Our journey begins by addressing the distribution of the term outside the parentheses, a crucial step in simplifying the equation. This distribution allows us to eliminate the parentheses and pave the way for further simplification. Next, we will strategically combine like terms, a fundamental technique in algebra that involves grouping terms with similar properties. This process streamlines the equation, making it more manageable and conducive to solving for x. Subsequently, we will employ the principle of inverse operations, a cornerstone of equation solving, to isolate x on one side of the equation. This involves performing operations that undo existing operations, effectively peeling away the layers surrounding x. Throughout this process, we will maintain a steadfast commitment to preserving the equality of the equation, ensuring that each operation performed on one side is mirrored on the other. This principle is the bedrock of equation solving, guaranteeing that the solution obtained is valid and accurate. By meticulously executing these steps, we will navigate the equation with precision, ultimately arriving at the value of x that satisfies the given condition. This step-by-step approach not only provides a clear pathway to the solution but also reinforces the importance of methodical problem-solving in mathematics.

1. Distribute the term $-\frac{5}{2}$ across the parentheses:

   We begin by distributing the term ${-\frac{5}{2}}$ across the parentheses, multiplying it by each term inside. This process eliminates the parentheses and simplifies the equation, making it easier to manipulate. The distribution is a fundamental algebraic technique that allows us to deal with expressions enclosed in parentheses, effectively breaking them down into simpler terms. This step is crucial in isolating the variable x and paving the way for further simplification.

   \begin{aligned} \frac{635}{72} &= -\frac{5}{2}\left(-\frac{11}{4}+x\right) \ \frac{635}{72} &= -\frac{5}{2} \cdot -\frac{11}{4} + -\frac{5}{2} \cdot x \ \frac{635}{72} &= \frac{55}{8} - \frac{5}{2}x \end{aligned}

2. Subtract $\frac{55}{8}$ from both sides:

   To isolate the term containing x, we subtract ${\frac{55}{8}}$ from both sides of the equation. This operation maintains the equality of the equation while moving us closer to isolating x. Subtracting the same value from both sides is a fundamental principle in equation solving, ensuring that the balance of the equation is preserved. This step is crucial in our quest to determine the value of x.

   \begin{aligned} \frac{635}{72} - \frac{55}{8} &= \frac{55}{8} - \frac{5}{2}x - \frac{55}{8} \ \frac{635}{72} - \frac{55 \cdot 9}{8 \cdot 9} &= -\frac{5}{2}x \ \frac{635}{72} - \frac{495}{72} &= -\frac{5}{2}x \ \frac{140}{72} &= -\frac{5}{2}x \end{aligned}

3. Simplify the fraction $\frac{140}{72}$:

   Before proceeding further, we simplify the fraction ${\frac{140}{72}}$ to its lowest terms. Simplifying fractions makes the equation more manageable and reduces the risk of errors in subsequent calculations. This step involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. Simplifying fractions is a fundamental skill in mathematics, essential for efficient problem-solving.

   \begin{aligned} \frac{140}{72} &= \frac{4 \cdot 35}{4 \cdot 18} \ \frac{140}{72} &= \frac{35}{18} \end{aligned}

   Now our equation looks like this:

   \frac{35}{18} = -\frac{5}{2}x

4. Multiply both sides by $-\frac{2}{5}$:

   To isolate x, we multiply both sides of the equation by the reciprocal of ${-\frac{5}{2}}*, which is *\${-\\frac{2}{5}\}. Multiplying by the reciprocal is a key technique in solving equations, as it effectively cancels out the coefficient of the variable. This operation maintains the equality of the equation while bringing us closer to the solution. Multiplying by the reciprocal is a fundamental tool in algebra, enabling us to isolate variables and solve equations efficiently.

   \begin{aligned} -\frac{2}{5} \cdot \frac{35}{18} &= -\frac{5}{2}x \cdot -\frac{2}{5} \ -\frac{2 \cdot 35}{5 \cdot 18} &= x \ -\frac{70}{90} &= x \end{aligned}

5. Simplify the fraction $-\frac{70}{90}$:

   Finally, we simplify the fraction ${-\frac{70}{90}}$ to its lowest terms. Simplifying fractions is a crucial step in presenting the solution in its most concise form. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. A simplified fraction is easier to interpret and use in further calculations.

   \begin{aligned} x &= -\frac{70}{90} \ x &= -\frac{10 \cdot 7}{10 \cdot 9} \ x &= -\frac{7}{9} \end{aligned}

The Solution: x = -\frac{7}{9}

Therefore, the solution to the equation ${\frac{635}{72}=-\frac{5}{2}\left(-\frac{11}{4}+x\right)}$ is x = ${-\frac{7}{9}}*. This solution represents the value of *x* that satisfies the given equation, making the left-hand side equal to the right-hand side. The solution is the culmination of our methodical algebraic manipulations, each step carefully executed to isolate *x* and determine its value. This result not only provides the answer to the problem but also validates the correctness of our solution process. The journey to this solution has reinforced our understanding of equation-solving techniques and the importance of precision in mathematical calculations. The value *x* = *\${-\\frac{7}{9}\} is the key that unlocks the equation, revealing the hidden relationship between the terms and providing a definitive answer.

Verification of the Solution

To ensure the accuracy of our solution, it is essential to verify that x = ${-\frac{7}{9}}* indeed satisfies the original equation. Verification is a critical step in the problem-solving process, providing a safeguard against errors and confirming the validity of the solution. This process involves substituting the obtained value of *x* back into the original equation and checking if both sides of the equation are equal. If the left-hand side (LHS) equals the right-hand side (RHS), then our solution is verified and we can be confident in its correctness. The verification process not only confirms the solution but also deepens our understanding of the equation and the relationship between its terms. It is a testament to the rigor of mathematical problem-solving, where every step is subject to scrutiny and validation. Let us now embark on the verification process, substituting *x* = *\${-\\frac{7}{9}\} into the original equation and meticulously evaluating both sides to ensure their equality.

Substituting x = ${-\frac{7}{9}}$ into the original equation:

\begin{aligned} \frac{635}{72} &= -\frac{5}{2}\left(-\frac{11}{4} + \left(-\frac{7}{9}\right)\right) \ \frac{635}{72} &= -\frac{5}{2}\left(-\frac{11}{4} - \frac{7}{9}\right) \ \frac{635}{72} &= -\frac{5}{2}\left(-\frac{11 \cdot 9}{4 \cdot 9} - \frac{7 \cdot 4}{9 \cdot 4}\right) \ \frac{635}{72} &= -\frac{5}{2}\left(-\frac{99}{36} - \frac{28}{36}\right) \ \frac{635}{72} &= -\frac{5}{2}\left(-\frac{127}{36}\right) \ \frac{635}{72} &= \frac{5 \cdot 127}{2 \cdot 36} \ \frac{635}{72} &= \frac{635}{72} \end{aligned}

Since the left-hand side (LHS) equals the right-hand side (RHS), our solution x = ${-\frac{7}{9}}$ is verified.

Conclusion: Mastering Algebraic Equations

In conclusion, we have successfully navigated the intricate landscape of the equation ${\frac{635}{72}=-\frac{5}{2}\left(-\frac{11}{4}+x\right)}, meticulously deciphering its complexities to arrive at the solution *x* = *\${-\\frac{7}{9}\}*. This journey has not only provided us with a numerical answer but has also illuminated the fundamental principles of equation solving, underscoring the importance of systematic algebraic manipulation. We have witnessed the power of distribution, the elegance of combining like terms, and the efficacy of inverse operations in isolating variables. The verification process has further solidified our confidence in the solution, demonstrating the rigor and precision inherent in mathematical problem-solving. This exploration serves as a testament to the transformative potential of algebraic equations, which are not merely abstract constructs but powerful tools for modeling and understanding the world around us. By mastering these techniques, we empower ourselves to tackle a wide range of mathematical challenges, from simple linear equations to complex scientific models. The ability to solve equations is a cornerstone of mathematical literacy, opening doors to further exploration and discovery in various fields of study. As we conclude this exploration, we carry with us not just the solution to this specific equation but also a deeper appreciation for the beauty and power of algebra.