Solving The Equation 4/x = 14/(5x) + 2 A Step-by-Step Guide

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In this article, we will delve into the process of solving the given equation: 4x=145x+2\frac{4}{x}=\frac{14}{5 x}+2. This equation involves fractions and a variable in the denominator, which means we need to be careful about potential values of $x$ that could make the denominator zero. We'll go through the steps required to isolate the variable and find its value. The equation presented, 4x=145x+2\frac{4}{x}=\frac{14}{5 x}+2, is a rational equation. Solving rational equations involves clearing the fractions and then solving the resulting algebraic equation. This process typically involves finding a common denominator, multiplying both sides of the equation by this common denominator, and then simplifying. It is crucial to check the solutions to avoid extraneous roots, which are solutions that arise from the solving process but do not satisfy the original equation. These can occur when multiplying both sides by an expression that can be zero. As we proceed, each step will be explained in detail, ensuring a clear understanding of the methodology employed. By carefully following each stage, you'll grasp the techniques required to tackle similar algebraic problems with confidence. Understanding these steps is crucial not only for solving equations in mathematics but also for various applications in physics, engineering, and other scientific fields where rational equations frequently arise. Solving this equation will allow us to improve our problem-solving skills and deepen our understanding of mathematical principles. In the following sections, we will break down each step, providing clarity and insights that build toward a comprehensive understanding of rational equation solutions.

Step-by-Step Solution

1. Identify the Domain

The first step in solving any equation, especially one involving fractions, is to identify the domain, or the set of all possible values for the variable. In the equation 4x=145x+2\frac{4}{x}=\frac{14}{5 x}+2, we have variables in the denominators of fractions. Specifically, we have $x$ and $5x$ in the denominators. To avoid division by zero, we must ensure that $x\neq 0$. Therefore, the domain of this equation is all real numbers except 0. This is a critical step, as any solution we find later must be within this domain; otherwise, it would be an extraneous solution. Recognizing and setting domain restrictions from the beginning helps in avoiding potential errors later in the process. Ignoring the domain may lead to accepting solutions that are not valid within the original context of the equation. Understanding domain restrictions is a foundational aspect of algebra, and it is particularly important when dealing with rational expressions and equations. The recognition that $x$ cannot be 0 ensures that the mathematical operations we perform remain valid and meaningful. Identifying the domain is a proactive step that significantly aids in simplifying the solution process and maintaining accuracy.

2. Find the Least Common Denominator (LCD)

The next step is to find the least common denominator (LCD) of the fractions in the equation. In the equation 4x=145x+2\frac{4}{x}=\frac{14}{5 x}+2, we have two fractions: $\frac{4}{x}$ and $\frac{14}{5x}$. To find the LCD, we look at the denominators, which are $x$ and $5x$. The LCD is the smallest expression that both denominators divide into evenly. In this case, the LCD is $5x$. This is because $5x$ is divisible by both $x$ and $5x$. Identifying the LCD is crucial because it allows us to eliminate the fractions from the equation, which simplifies the process of solving for $x$. The concept of LCD is not just limited to solving algebraic equations; it's also fundamental in adding and subtracting fractions in arithmetic. Mastery of finding the LCD is an essential skill for more advanced mathematical topics like calculus, where manipulation of complex rational functions is commonplace. Correctly determining the LCD ensures the subsequent algebraic manipulations are valid and lead to accurate solutions. This step reduces complexity and sets the stage for solving the equation more effectively.

3. Multiply Both Sides by the LCD

Now that we have the least common denominator (LCD), we multiply both sides of the equation 4x=145x+2\frac{4}{x}=\frac{14}{5 x}+2$ by the LCD, which is $5x$. This is a crucial step because it clears the fractions from the equation, making it easier to solve. When we multiply both sides by $5x$, we get:

5x×4x=5x×(145x+2){ 5x \times \frac{4}{x} = 5x \times \left(\frac{14}{5 x}+2\right) }

This simplifies to:

5×4=14+5x×2{ 5 \times 4 = 14 + 5x \times 2 }

Further simplification yields:

20=14+10x{ 20 = 14 + 10x }

Multiplying both sides of an equation by the same non-zero expression is a fundamental algebraic technique. This operation preserves the equality as long as we account for any restrictions on the variable (as we did when identifying the domain). The goal here is to transform the equation into a more manageable form, typically a linear equation, which is straightforward to solve. This step significantly reduces the complexity of the problem by eliminating denominators, thereby allowing us to apply basic algebraic manipulations more easily. The multiplication must be performed carefully, ensuring each term on both sides is correctly multiplied by the LCD. This transformation is a key strategy in solving various types of equations involving rational expressions.

4. Simplify and Solve for x

After multiplying both sides of the equation by the LCD, we have a simplified equation: $20 = 14 + 10x$. The next step is to isolate the variable $x$ and solve for its value. First, we subtract 14 from both sides of the equation:

20−14=14+10x−14{ 20 - 14 = 14 + 10x - 14 }

This simplifies to:

6=10x{ 6 = 10x }

Now, to isolate $x$, we divide both sides by 10:

610=10x10{ \frac{6}{10} = \frac{10x}{10} }

This gives us:

x=610{ x = \frac{6}{10} }

We can further simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

x=35{ x = \frac{3}{5} }

Solving for $x$ typically involves a sequence of algebraic manipulations designed to isolate the variable on one side of the equation. Each operation must be applied equally to both sides to maintain the balance of the equation. The goal is to progressively simplify the equation until the variable stands alone, revealing its value. This process often involves combining like terms, using inverse operations (such as addition and subtraction, or multiplication and division), and factoring or expanding expressions as needed. The ability to solve equations is a fundamental skill in mathematics and is applied across numerous disciplines and real-world scenarios.

5. Check the Solution

The final and arguably one of the most crucial steps in solving an equation is to check the solution. We found that $x = \frac{3}{5}∗∗isapotentialsolutiontotheequation∗∗** is a potential solution to the equation **\frac{4}{x}=\frac{14}{5 x}+2$. To verify this, we substitute this value back into the original equation:

435=145×35+2{ \frac{4}{\frac{3}{5}} = \frac{14}{5 \times \frac{3}{5}} + 2 }

Simplifying the left side of the equation, we have:

435=4×53=203{ \frac{4}{\frac{3}{5}} = 4 \times \frac{5}{3} = \frac{20}{3} }

Simplifying the right side of the equation, we have:

145×35+2=143+2{ \frac{14}{5 \times \frac{3}{5}} + 2 = \frac{14}{3} + 2 }

To add the terms on the right side, we need a common denominator, which is 3:

143+2=143+63=203{ \frac{14}{3} + 2 = \frac{14}{3} + \frac{6}{3} = \frac{20}{3} }

Since the left side equals the right side ($\frac{20}{3} = \frac{20}{3}),oursolution∗∗$x=35), our solution **\$x = \frac{3}{5}** is correct. Checking solutions is essential because it helps identify any extraneous solutions that may have arisen due to the process of solving the equation, especially when dealing with rational or radical equations. Extraneous solutions are values that satisfy a transformed equation but not the original equation. This step confirms that the solution is valid within the context of the initial problem.

Conclusion

In conclusion, we have successfully solved the equation 4x=145x+2\frac{4}{x}=\frac{14}{5 x}+2$. By following a step-by-step approach, we first identified the domain to ensure we avoided any division by zero. We then found the least common denominator (LCD), multiplied both sides of the equation by the LCD to eliminate fractions, simplified the equation, and solved for $x$. Finally, we checked our solution to ensure its validity. The solution to the equation is $x = \frac{3}{5}$. This exercise demonstrates the importance of methodical problem-solving in algebra. Each step, from identifying domain restrictions to checking the solution, plays a crucial role in arriving at the correct answer. The techniques used here, such as clearing fractions and isolating variables, are fundamental to solving a wide range of algebraic equations. Mastering these techniques not only improves mathematical skills but also enhances problem-solving abilities applicable in various fields. The process of solving this equation reinforces the idea that mathematics is about logical progression and attention to detail. By systematically applying algebraic principles, we can confidently tackle complex problems and find accurate solutions.