Solve The Equation \(\frac{12}{14x-5} = \frac{4}{5x}\).

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Introduction

In this article, we will delve into the process of solving for the variable x in the equation ${\frac{12}{14x-5} = \frac{4}{5x}}$. This type of equation, involving fractions with algebraic expressions in the denominators, is common in algebra and requires a systematic approach to find the solution. Understanding how to solve such equations is a fundamental skill in mathematics, with applications in various fields such as physics, engineering, and economics. This article provides a comprehensive, step-by-step guide to tackling this problem, ensuring clarity and ease of understanding for students and math enthusiasts alike. Solving for x involves algebraic manipulation, cross-multiplication, and potentially solving a quadratic equation. Let's embark on this mathematical journey to demystify the solution process.

Understanding the Equation

The equation presented, ${\frac{12}{14x-5} = \frac{4}{5x}}$, involves two rational expressions set equal to each other. To effectively solve for x, we must eliminate the fractions. The first critical step is to identify the restrictions on x. Since division by zero is undefined, we need to determine the values of x that would make either denominator equal to zero. These values must be excluded from our final solution set. Specifically, we need to ensure that ${14x - 5 \neq 0}$ and ${5x \neq 0}$. Solving these inequalities will give us the restricted values of x. Once we've identified these restrictions, we can proceed with the algebraic manipulation to solve for x. The next step typically involves cross-multiplication, a technique that simplifies the equation by removing the fractions. This involves multiplying the numerator of the first fraction by the denominator of the second, and vice versa. This process will transform the equation into a more manageable form, which we can then solve using standard algebraic techniques. By carefully considering these initial steps, we lay a solid foundation for finding the correct solution.

Step-by-Step Solution

1. Identify Restrictions on x

To begin solving for x, the first crucial step is to determine any values of x that would make the denominators of the fractions equal to zero. These values are restricted because division by zero is undefined in mathematics. For the given equation, ${\frac{12}{14x-5} = \frac{4}{5x}}$, we have two denominators: ${14x - 5}$ and ${5x}$. We need to find the values of x for which these denominators are not equal to zero.

a. For the denominator ${14x - 5}$

We set ${14x - 5 \neq 0}$ and solve for x:

${ 14x - 5 \neq 0 }$

${ 14x \neq 5 }$

${ x \neq \frac{5}{14} }$

So, x cannot be equal to ${\frac{5}{14}}$.

b. For the denominator ${5x}$

We set ${5x \neq 0}$ and solve for x:

${ 5x \neq 0 }$

${ x \neq 0 }$

Thus, x cannot be equal to 0.

Therefore, the restrictions on x are ${x \neq \frac{5}{14}}$ and ${x \neq 0}$. These values must be excluded from the final solution set.

2. Cross-Multiplication

Once we have identified the restrictions on x, the next step is to eliminate the fractions by cross-multiplication. This technique involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. In our equation, ${\frac{12}{14x-5} = \frac{4}{5x}}$, we will multiply 12 by ${5x}$ and 4 by ${(14x - 5)}$.

This gives us:

${ 12 \cdot 5x = 4 \cdot (14x - 5) }$

Simplifying both sides of the equation, we get:

${ 60x = 4(14x - 5) }$

3. Distribute and Simplify

Now, we need to distribute the 4 on the right side of the equation:

${ 60x = 4 \cdot 14x - 4 \cdot 5 }$

${ 60x = 56x - 20 }$

4. Isolate the Variable

To isolate the variable x, we need to get all the x terms on one side of the equation and the constants on the other side. We can subtract ${56x}$ from both sides:

${ 60x - 56x = 56x - 20 - 56x }$

${ 4x = -20 }$

5. Solve for x

Finally, to solve for x, we divide both sides of the equation by 4:

${ \frac{4x}{4} = \frac{-20}{4} }$

${ x = -5 }$

Verification

6. Check the Solution

After finding a potential solution for x, it is crucial to verify that this solution is valid by substituting it back into the original equation. This step ensures that our solution does not lead to any undefined operations, such as division by zero, and that both sides of the equation remain equal.

Our solution is ${x = -5}$. Let's substitute this value into the original equation:

${ \frac{12}{14x-5} = \frac{4}{5x} }$

Substituting ${x = -5}$, we get:

${ \frac{12}{14(-5)-5} = \frac{4}{5(-5)} }$

Now, we simplify both sides:

${ \frac{12}{-70-5} = \frac{4}{-25} }$

${ \frac{12}{-75} = \frac{4}{-25} }$

Simplify the fractions by dividing both the numerator and the denominator by their greatest common divisor. For the left side, the greatest common divisor of 12 and 75 is 3. For the right side, the greatest common divisor of 4 and 25 is 1 (they are already in simplest form):

${ \frac{12 \div 3}{-75 \div 3} = \frac{4}{-25} }$

${ \frac{4}{-25} = \frac{4}{-25} }$

Since both sides of the equation are equal, our solution ${x = -5}$ is valid.

7. Check against Restrictions

We must also ensure that our solution does not violate any of the restrictions we identified earlier. We found that ${x}$ cannot be equal to ${\frac{5}{14}}$ or 0. Our solution, ${x = -5}$, does not fall into these restricted values, so it is a valid solution.

Conclusion

In conclusion, we have successfully solved for x in the equation ${\frac{12}{14x-5} = \frac{4}{5x}}$. By following a step-by-step approach, we first identified the restrictions on x to ensure we avoid division by zero. Then, we used cross-multiplication to eliminate the fractions, simplified the equation, and isolated the variable x. Finally, we verified our solution by substituting it back into the original equation and checking it against the restrictions. The solution we found is ${x = -5}$. This process demonstrates the importance of careful algebraic manipulation and verification in solving equations involving rational expressions. Understanding these steps is crucial for success in algebra and related mathematical fields. Remember to always check for restrictions and verify your solutions to ensure accuracy. The techniques used in this article can be applied to a variety of similar problems, reinforcing the fundamental principles of algebraic problem-solving.