Solving The Equation (2x+7)/(5x+8) = (2x+6)/(5x+4) Step-by-Step
Solving algebraic equations is a fundamental skill in mathematics. In this comprehensive guide, we will delve into the step-by-step solution of the equation . This involves cross-multiplication, simplification, and ultimately isolating x to find its value. Understanding these techniques is essential for anyone studying algebra or related fields. Our focus will be on providing a clear, detailed explanation that not only solves the equation but also reinforces the underlying algebraic principles. We aim to make this process accessible and understandable, even for those who may be new to this type of problem.
The journey to finding the value of x begins with recognizing the structure of the equation. We are presented with two rational expressions set equal to each other. This structure immediately suggests the use of cross-multiplication as a primary strategy. Cross-multiplication allows us to eliminate the fractions and transform the equation into a more manageable linear form. The subsequent steps involve carefully distributing terms, combining like terms, and isolating x on one side of the equation. Each step requires precision and a solid grasp of algebraic manipulations. We will emphasize the importance of maintaining balance in the equation and performing the same operations on both sides to ensure the solution's validity. Throughout this explanation, we will also highlight potential pitfalls and common mistakes to help readers avoid them. This holistic approach aims to enhance the understanding and proficiency in solving similar algebraic problems.
Step-by-Step Solution
To solve the equation , we will proceed with the following steps:
1. Cross-Multiplication
Our initial step in solving this equation involves cross-multiplication. This technique is crucial for eliminating the fractions and transforming the equation into a more manageable form. By multiplying the numerator of the first fraction by the denominator of the second and vice versa, we can effectively clear the denominators. This process simplifies the equation, making it easier to solve for the unknown variable x. The accuracy of this step is paramount, as any error here will propagate through the rest of the solution. Therefore, we will carefully detail each multiplication to ensure clarity and precision. This step sets the foundation for the subsequent algebraic manipulations that will lead us to the solution.
Multiply both sides by (5x + 8)(5x + 4):
This step eliminates the fractions, making the equation easier to work with. The equation is now in a form where we can apply the distributive property and simplify further. This transformation is a key step in solving equations involving rational expressions, as it allows us to transition from a complex fractional equation to a more familiar polynomial equation. By carefully executing this cross-multiplication, we set the stage for the subsequent steps of expanding, simplifying, and isolating the variable x. The clarity and accuracy of this initial step are crucial for arriving at the correct solution.
2. Expansion
Following the cross-multiplication, the next critical step is to expand both sides of the equation. This involves applying the distributive property, often referred to as the FOIL (First, Outer, Inner, Last) method, to multiply the binomial expressions. The expansion process is essential for removing the parentheses and combining like terms, which will eventually lead us to isolating the variable x. Accuracy in this step is paramount, as any error in the multiplication or sign handling can lead to an incorrect solution. Therefore, we will carefully break down each term-by-term multiplication to ensure clarity and precision. This meticulous approach will help prevent common mistakes and ensure a solid foundation for the subsequent simplification steps.
Expand both sides:
This expansion is achieved by multiplying each term in the first binomial by each term in the second binomial. On the left side, (2x + 7) is multiplied by (5x + 4), resulting in the terms 10x², 8x, 35x, and 28. Similarly, on the right side, (2x + 6) is multiplied by (5x + 8), yielding 10x², 16x, 30x, and 48. This process effectively removes the parentheses and transforms the equation into a polynomial form, setting the stage for the next step of combining like terms and simplifying.
3. Simplification
After expanding both sides of the equation, the next crucial step is simplification. This involves combining like terms on each side to consolidate the expression and make it more manageable. Simplification is essential for reducing the complexity of the equation and paving the way for isolating the variable x. The accuracy of this step is vital, as any error in combining terms will propagate through the rest of the solution. We will carefully identify and combine like terms, paying close attention to the signs and coefficients. This meticulous approach ensures that the equation is simplified correctly, leading us closer to the final solution. By simplifying effectively, we reduce the equation to its most basic form, making the subsequent steps of solving for x more straightforward.
Combine like terms:
Here, on the left side, the 8x and 35x terms are combined to give 43x. Similarly, on the right side, 16x and 30x are combined to give 46x. The equation now presents a clearer picture of the relationship between the terms, making it easier to proceed with the process of isolating the variable x. This simplification step is a critical part of solving algebraic equations, as it streamlines the expression and reduces the likelihood of errors in subsequent manipulations. By accurately combining like terms, we move closer to the final solution.
4. Isolating x
With the equation simplified, the next strategic step is to isolate the variable x on one side. This process involves performing algebraic operations to cancel out terms and group the x terms together. Isolating x is a fundamental technique in solving equations, as it directly leads to the solution. The key to this step is to maintain balance in the equation by performing the same operations on both sides. This ensures that the equality is preserved and the solution remains valid. We will carefully execute each step, ensuring that x is isolated without altering the equation's fundamental properties. This systematic approach will lead us to the value of x that satisfies the original equation.
Subtract from both sides:
This eliminates the quadratic term, simplifying the equation further and making it a linear equation in terms of x. The equal subtraction of 10x² from both sides maintains the equation's balance, ensuring that the solution remains unchanged. This step is crucial for reducing the complexity of the equation and setting the stage for isolating the variable x. By removing the quadratic term, we can now focus on the linear terms and proceed with the algebraic manipulations necessary to solve for x.
Subtract 43x from both sides:
By subtracting 43x from both sides, we effectively group the x terms on one side of the equation. This is a pivotal step in isolating x, as it brings us closer to determining its value. The subtraction maintains the equation's balance, ensuring that the equality remains valid. This manipulation is a fundamental technique in solving algebraic equations, allowing us to separate the variable from the constants. With the x term now isolated on one side, we can proceed with the remaining steps to find the numerical value of x.
Subtract 48 from both sides:
This step isolates the term with x on one side of the equation. By subtracting 48 from both sides, we move closer to solving for x. This operation maintains the equation's balance, ensuring the equality remains true. Isolating the x term is a critical technique in algebra, as it allows us to directly determine the value of the variable. With this step completed, we are now in the final stage of solving for x, which involves dividing both sides by the coefficient of x.
Divide both sides by 3:
x = - rac{20}{3}
This final step gives us the value of x. By dividing both sides by 3, we isolate x and find that it equals -20/3. This operation completes the process of solving the equation, providing us with the solution. The division maintains the equation's balance, ensuring the equality holds true. Finding the value of x is the ultimate goal of solving algebraic equations, and this step achieves that. The solution x = -20/3 satisfies the original equation, making it the correct answer.
5. Convert to Mixed Number
Our final step involves converting the improper fraction to a mixed number for clarity and ease of interpretation. Converting improper fractions to mixed numbers is a common practice in mathematics, as it presents the value in a more intuitive form. This step doesn't change the value of x but expresses it in a way that is often easier to understand and compare. The process involves dividing the numerator by the denominator and expressing the result as a whole number and a fraction. This conversion enhances the readability of the solution and provides a more comprehensive understanding of the value of x. By expressing the solution as a mixed number, we make it more accessible and relatable to a broader audience.
Convert to a mixed number:
x = -6 rac{2}{3}
This conversion is done by dividing 20 by 3, which gives a quotient of 6 and a remainder of 2. The mixed number is then written as -6 and 2/3. This final representation of the solution provides a clearer understanding of the value of x, as it expresses the number as a whole number and a proper fraction. The conversion to a mixed number is a standard practice in mathematics, making the solution more accessible and easier to interpret. The final answer, x = -6 2/3, is the solution to the original equation, expressed in a clear and concise format.
Final Answer
Therefore, the solution to the equation is:
This final answer represents the value of x that satisfies the original equation . Through a series of algebraic manipulations, including cross-multiplication, expansion, simplification, and isolation of x, we have arrived at this solution. The answer is presented in a mixed number format for clarity and ease of understanding. This solution is the culmination of a step-by-step process that demonstrates the application of fundamental algebraic principles. The accuracy and precision employed in each step ensure the validity of this result. This final answer serves as a clear and concise resolution to the problem, showcasing the power of algebraic techniques in solving equations.
Conclusion
In conclusion, we have successfully solved the equation and found the value of x to be -6 2/3. This process involved a series of algebraic steps, each requiring careful execution and a solid understanding of mathematical principles. From the initial cross-multiplication to the final conversion into a mixed number, each step played a crucial role in arriving at the solution. The methodical approach used in this explanation highlights the importance of precision and attention to detail in solving algebraic equations. This comprehensive solution not only provides the answer but also reinforces the underlying concepts and techniques involved. By understanding these methods, one can confidently tackle similar algebraic problems and enhance their mathematical skills.
The journey through solving this equation underscores the interconnectedness of algebraic concepts. The application of cross-multiplication, expansion, simplification, and isolation of variables are all essential tools in the mathematician's toolkit. The ability to seamlessly transition between these techniques is key to solving a wide range of algebraic problems. This solution serves as a practical example of how these techniques can be applied in a step-by-step manner to arrive at a definitive answer. The emphasis on clarity and precision throughout the process aims to demystify the process of equation-solving and empower readers to approach mathematical challenges with confidence. This exploration not only solves a specific equation but also illuminates the broader landscape of algebraic problem-solving.