Solving For X In 3/(x-4) = 7/x A Step By Step Guide

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Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts. In this comprehensive guide, we will delve into the process of solving for x in the equation 3/(x-4) = 7/x. This particular equation involves fractions and variables in the denominator, requiring a systematic approach to arrive at the correct solution. Understanding the steps involved in solving this equation will not only enhance your algebraic proficiency but also provide a framework for tackling similar problems in the future.

Understanding the Equation and the Goal

The equation 3/(x-4) = 7/x is a rational equation, which means it involves fractions where the numerator and/or denominator contain variables. Our primary goal is to isolate x on one side of the equation to determine its value. However, before we jump into the solving process, it's crucial to identify any restrictions on the variable. In this case, the denominators cannot be equal to zero, as division by zero is undefined. Therefore, x cannot be equal to 4 or 0.

Identifying Restrictions:

  • x - 4 ≠ 0 => x ≠ 4
  • x ≠ 0

These restrictions are essential because any solution we find must satisfy these conditions. If we obtain a value for x that violates these restrictions, it is considered an extraneous solution and must be discarded.

Step-by-Step Solution

Now that we understand the equation and its restrictions, let's proceed with the step-by-step solution:

Step 1: Cross-Multiplication

The first step in solving the equation is to eliminate the fractions. We can achieve this by cross-multiplying. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. In our equation, this translates to:

3 * x = 7 * (x - 4)

This step effectively clears the denominators and transforms the equation into a more manageable form.

Step 2: Distribute and Simplify

Next, we need to distribute the 7 on the right side of the equation:

3x = 7x - 28

Now, we have a linear equation that we can solve by isolating x. To do this, we need to group the x terms on one side of the equation and the constants on the other side.

Step 3: Group Like Terms

Subtract 7x from both sides of the equation to get all the x terms on the left side:

3x - 7x = -28

This simplifies to:

-4x = -28

Step 4: Isolate x

To isolate x, divide both sides of the equation by -4:

x = -28 / -4

This gives us:

x = 7

Step 5: Check for Extraneous Solutions

Before we declare our solution, we must check if it violates any restrictions we identified earlier. We found that x cannot be equal to 4 or 0. Our solution, x = 7, does not violate these restrictions. Therefore, it is a valid solution.

Verifying the Solution

To ensure our solution is correct, we can substitute x = 7 back into the original equation:

3 / (7 - 4) = 7 / 7

3 / 3 = 1

1 = 1

Since the equation holds true, our solution x = 7 is correct.

Alternative Methods and Considerations

While cross-multiplication is a common method for solving equations involving fractions, there are alternative approaches that can be used. One such method is finding a common denominator. Let's briefly explore this approach:

Method 2: Finding a Common Denominator

To solve the equation using a common denominator, we need to find the least common multiple (LCM) of the denominators, which in this case is x(x - 4). Multiply both sides of the equation by the LCM:

[x(x - 4)] * [3 / (x - 4)] = [x(x - 4)] * [7 / x]

This simplifies to:

3x = 7(x - 4)

Notice that we arrive at the same equation as in Step 2 of the cross-multiplication method. From here, we can proceed with the same steps to solve for x.

Considerations for Complex Equations

In more complex equations, finding a common denominator might be a more efficient approach, especially when dealing with multiple fractions or more complicated denominators. However, for simple equations like the one we solved, cross-multiplication is often a quicker method.

Common Mistakes to Avoid

When solving rational equations, it's crucial to avoid common mistakes that can lead to incorrect solutions. Here are some pitfalls to watch out for:

  1. Forgetting to Check for Restrictions: Failing to identify and check for restrictions on the variable can lead to extraneous solutions. Always determine the values that make the denominators zero and ensure your solution does not match these values.
  2. Incorrect Cross-Multiplication: Make sure to multiply the numerator of each fraction by the denominator of the opposite fraction. A simple mistake in this step can invalidate the entire solution.
  3. Distribution Errors: When distributing, ensure that you multiply the constant by every term inside the parentheses. For example, in the equation 3x = 7(x - 4), the 7 must be multiplied by both x and -4.
  4. Sign Errors: Pay close attention to signs when grouping like terms and performing operations. A sign error can easily lead to an incorrect solution.

Practice Problems

To solidify your understanding, let's work through a few practice problems:

Practice Problem 1:

Solve for x: 5 / (x + 2) = 2 / x

Solution:

  1. Cross-multiply: 5x = 2(x + 2)
  2. Distribute: 5x = 2x + 4
  3. Group like terms: 3x = 4
  4. Isolate x: x = 4 / 3
  5. Check for restrictions: x ≠ -2, x ≠ 0. The solution x = 4/3 is valid.

Practice Problem 2:

Solve for x: (x + 1) / 4 = (2x - 1) / 3

Solution:

  1. Cross-multiply: 3(x + 1) = 4(2x - 1)
  2. Distribute: 3x + 3 = 8x - 4
  3. Group like terms: -5x = -7
  4. Isolate x: x = 7 / 5
  5. Check for restrictions: There are no restrictions in this case, as the denominators are constants. The solution x = 7/5 is valid.

Real-World Applications

Solving rational equations is not just an abstract mathematical exercise; it has practical applications in various fields. Here are a couple of examples:

  1. Physics: Rational equations are used in physics to describe relationships between quantities such as speed, distance, and time. For example, the equation speed = distance / time is a rational equation.
  2. Engineering: Engineers use rational equations in circuit analysis, fluid dynamics, and other areas. For instance, the current in an electrical circuit can be calculated using rational equations based on Ohm's Law.
  3. Economics: Economists use rational equations to model supply and demand curves, as well as other economic relationships.

By mastering the skill of solving rational equations, you'll be better equipped to tackle problems in these and other fields.

Conclusion

In this guide, we've provided a comprehensive walkthrough of solving for x in the equation 3/(x-4) = 7/x. We've covered the importance of identifying restrictions, the step-by-step solution process, alternative methods, common mistakes to avoid, practice problems, and real-world applications. By understanding and applying these concepts, you'll be well-prepared to solve a wide range of rational equations. Remember, practice is key to mastering any mathematical skill, so keep working through problems and refining your understanding. With dedication and perseverance, you'll become proficient in solving rational equations and unlock new possibilities in mathematics and beyond.