Solving Equations Step By Step A Guide To X = (4x - 24)/3 = 4x/5

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In this article, we will walk through the process of solving the equation x = (4x - 24)/3 = 4x/5. This type of equation, which involves fractions and variables, requires careful manipulation to isolate the variable x. Our goal is to find the value of x that satisfies the equation. If no such value exists, we will determine that there is NO SOLUTION. This article provides a comprehensive guide to solving this mathematical problem, including step-by-step instructions and explanations. We aim to make the solution process clear and understandable for anyone who wants to improve their algebra skills. Solving this equation involves several algebraic manipulations, such as clearing fractions, combining like terms, and isolating the variable. Each step is crucial in arriving at the correct solution. If you are new to solving equations, this guide will walk you through each step, ensuring that you understand the underlying principles. By the end of this article, you will have a clear understanding of how to approach and solve this type of equation. Understanding the methods to solve this equation not only helps in academic settings but also enhances problem-solving skills applicable in various real-life scenarios. Mathematical proficiency is a valuable asset, and mastering algebraic equations is a key component of that proficiency. This equation presents an interesting challenge, and we are here to guide you through every aspect of it, ensuring that you can solve similar problems confidently in the future.

1. Clearing the Fractions

To begin, we need to eliminate the fractions in the equation. The given equation is x = (4x - 24)/3 = 4x/5. To clear the fractions, we will multiply all parts of the equation by the least common multiple (LCM) of the denominators, which are 3 and 5. The LCM of 3 and 5 is 15. So, we will multiply each term by 15. This is a crucial step because dealing with fractions can be complex, and clearing them makes the equation easier to handle. Clearing fractions is a fundamental technique in solving algebraic equations, particularly when dealing with rational expressions. By multiplying through by the LCM, we ensure that all denominators are canceled out, leaving us with a simpler equation to solve. This method is based on the principle that multiplying both sides of an equation by the same non-zero number maintains the equality. The LCM method is widely used in mathematics to simplify equations involving fractions. It’s an efficient approach that minimizes the chances of errors and makes the subsequent steps more manageable. In this specific problem, multiplying by 15 will transform the equation into a form where we can easily isolate the variable x and find its value. This technique not only simplifies the equation but also sets the stage for further algebraic manipulations, such as combining like terms and isolating the variable. The elimination of fractions is a significant first step, making the problem more approachable and solvable. Without this step, the complexity of the fractions might obscure the underlying structure of the equation and make it harder to find the solution. Multiplying by the LCM is a foundational skill in algebra and essential for solving a wide range of equations.

1.1. Multiplying by 15

Multiplying each part of the equation x = (4x - 24)/3 = 4x/5 by 15, we get:

  • 15 * x = 15 * (4x - 24)/3 = 15 * (4x/5)

This simplifies to:

  • 15x = 5(4x - 24) = 3(4x)

This step effectively removes the fractions from the equation, making it easier to manage. Multiplying each term by 15 ensures that the equation remains balanced while eliminating the denominators. The result is a linear equation without fractions, which is more straightforward to solve. This transformation is crucial because fractions can complicate algebraic manipulations. By removing them, we simplify the equation and make it more accessible. The multiplication is done carefully, distributing 15 across each term and ensuring that all parts of the equation are affected equally. This meticulous approach is essential in mathematics to maintain accuracy and avoid errors. The resulting equation, 15x = 5(4x - 24) = 3(4x), is now ready for the next phase of solving, which involves expanding the terms and combining like terms. This process of multiplication by the LCM is a standard technique in algebra and provides a solid foundation for solving more complex equations. The elimination of fractions sets the stage for further simplification and makes the solution more attainable. The focus now shifts to working with whole numbers, which simplifies the subsequent algebraic steps. This methodical approach ensures that the solution process is clear and the chances of error are minimized. By applying this technique, we maintain the equation’s integrity while making it more manageable.

2. Expanding and Simplifying

Now we expand and simplify the equation 15x = 5(4x - 24) = 3(4x). This involves distributing the constants and combining like terms to streamline the equation. Expansion and simplification are crucial steps in solving algebraic equations, as they help to reveal the underlying structure of the equation and make it easier to solve. These steps involve applying the distributive property and combining terms that have the same variable or are constants. By simplifying the equation, we reduce the number of terms and make it more manageable. This process often reveals the relationship between the variables and constants, which is essential for finding the solution. Simplifying expressions is a fundamental skill in algebra and is used extensively in solving various types of equations. This step lays the groundwork for isolating the variable and finding its value. The goal is to transform the equation into its simplest form while maintaining its equality. Each operation is performed carefully to avoid errors and ensure that the simplified equation is equivalent to the original one. Simplification not only makes the equation easier to solve but also enhances understanding of the equation’s structure. This method is applicable across a wide range of algebraic problems, making it an invaluable technique in mathematics. The expansion and simplification process is a cornerstone of algebraic manipulation and is essential for tackling complex equations. Without simplification, equations can become unwieldy and challenging to solve, so mastering these techniques is critical for success in algebra.

2.1. Distributing the Constants

First, we distribute the constants in the equation:

  • 15x = 5(4x - 24) = 3(4x)

Expanding the terms, we get:

  • 15x = 20x - 120 = 12x

This step involves multiplying the constants outside the parentheses with each term inside the parentheses. The distributive property, which states that a(b + c) = ab + ac, is applied here to expand the expressions. By distributing the constants, we eliminate the parentheses and create a linear equation with individual terms. This expansion is crucial because it allows us to combine like terms and further simplify the equation. The process is methodical, ensuring that each term is correctly multiplied and that the equation remains balanced. This step is essential for making the equation more manageable and preparing it for subsequent simplification steps. The accuracy in distribution is critical to avoid errors that could affect the final solution. Careful attention to detail ensures that the expanded equation accurately reflects the original equation. Distributing constants is a foundational skill in algebra and is frequently used in simplifying various types of algebraic expressions and equations. This process is a key component of algebraic manipulation and helps to streamline the solution process. By expanding the terms, we move closer to isolating the variable and finding its value. This step also reveals the relationships between the different terms in the equation, making it easier to identify potential solutions.

2.2. Simplifying the Equation

Now, we have the equation 15x = 20x - 120 = 12x. We can split this into two separate equations:

  1. 15x = 20x - 120
  2. 20x - 120 = 12x

This separation allows us to focus on solving each equation independently. Splitting the equation into two parts simplifies the process by isolating different sets of terms. This approach breaks the problem into more manageable segments, making it easier to identify and apply appropriate solving techniques. The separation is based on the principle that if three expressions are equal (A = B = C), then A = B and B = C. This division is a strategic step that aids in isolating the variable and finding its value. Each equation can now be solved using standard algebraic methods, such as moving terms around and combining like terms. This technique is particularly useful when dealing with complex equations, as it allows for a more focused approach. Splitting the equation does not alter the solution; it merely restructures the problem to make it more solvable. This method highlights the importance of breaking down complex problems into simpler parts, a strategy that is widely applicable in both mathematics and other areas of problem-solving. The separated equations provide a clear path toward finding the solution by allowing for step-by-step manipulation of terms. This approach enhances understanding and reduces the likelihood of errors, making the overall solution process more efficient. By dividing the equation, we set the stage for isolating the variable in each part, ultimately leading to the determination of the solution or the conclusion that no solution exists.

3. Solving the Equations

3.1. Solving 15x = 20x - 120

To solve the equation 15x = 20x - 120, we need to isolate the variable x. First, we subtract 20x from both sides of the equation:

  • 15x - 20x = 20x - 120 - 20x

This simplifies to:

  • -5x = -120

Next, we divide both sides by -5:

  • -5x / -5 = -120 / -5

This gives us:

  • x = 24

3.2. Solving 20x - 120 = 12x

To solve the equation 20x - 120 = 12x, we again isolate the variable x. First, we subtract 12x from both sides:

  • 20x - 120 - 12x = 12x - 12x

This simplifies to:

  • 8x - 120 = 0

Next, we add 120 to both sides:

  • 8x - 120 + 120 = 0 + 120

This gives us:

  • 8x = 120

Finally, we divide both sides by 8:

  • 8x / 8 = 120 / 8

This results in:

  • x = 15

4. Checking for Consistency

We have obtained two different values for x from the two equations: x = 24 and x = 15. For the original equation to hold true, both values of x must be the same. Since 24 and 15 are different, there is NO SOLUTION to the original equation. Checking for consistency is a critical step in solving systems of equations. When an equation is split into multiple parts, each part must yield the same solution for the variable to ensure the overall equation is valid. In this case, the different values of x indicate that the original equation is inconsistent. Inconsistency arises when the equations derived from the original equation contradict each other, meaning there is no value of x that can satisfy all parts simultaneously. This step is not just a formality; it is a fundamental aspect of mathematical problem-solving. It helps in verifying the correctness of the solution and ensures that the answer is meaningful within the context of the original problem. In practical terms, consistency checks prevent the acceptance of spurious solutions, which are values that appear to solve an equation but do not satisfy the original conditions. This process reinforces the importance of rigorous mathematical thinking and attention to detail. By systematically verifying the results, we ensure the validity and reliability of the solution. This approach is widely applicable in various mathematical contexts and is a hallmark of thorough and accurate problem-solving. The conclusion that there is no solution highlights the significance of the consistency check and demonstrates that mathematical solutions must adhere to the fundamental rules of equality.

5. Conclusion

Since the values of x obtained from the two equations are different, there is NO SOLUTION to the original equation x = (4x - 24)/3 = 4x/5. This conclusion is reached after carefully manipulating and solving the equations derived from the original problem. The process involved clearing fractions, expanding terms, and isolating the variable x in two separate equations. By obtaining different values for x, we determined that no single value satisfies the entire equation. This outcome underscores the importance of verifying solutions and checking for consistency when dealing with multiple equations or conditions. Inconsistent equations like this one have no solution because the mathematical statements they represent contradict each other. Recognizing and identifying such cases is a key skill in algebra and mathematics in general. It prevents incorrect assumptions and ensures that the answers obtained are mathematically valid. The absence of a solution does not mean the problem is invalid; rather, it indicates that the given equation does not have a value for x that makes it true. This result is a valuable insight in itself, as it clarifies the nature of the equation and its limitations. Understanding how to determine that no solution exists is as important as knowing how to find a solution. This process reinforces logical reasoning and the application of algebraic principles to real-world problems. The detailed approach to solving this equation, including the consistency check, illustrates best practices in mathematical problem-solving and reinforces the significance of accuracy and precision in algebraic manipulations.