Solving 7 = 2/(x-9) = 9/(x+12) A Step-by-Step Guide

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Introduction

This article delves into the step-by-step process of solving the equation 7 = 2/(x-9) = 9/(x+12). This equation, which falls under the category of rational equations, requires a methodical approach to isolate the variable 'x' and determine its value. Rational equations, characterized by fractions with variables in the denominator, often present unique challenges due to the potential for extraneous solutions. Therefore, a thorough understanding of algebraic principles and careful execution of each step are essential for arriving at the correct solution. We will explore the concepts involved, the techniques applied, and the crucial steps necessary to verify the solution. This exploration will equip you with the skills to tackle similar rational equations confidently.

Understanding Rational Equations

Rational equations, like the one we're tackling, are equations that contain one or more fractions where the variable appears in the denominator. Solving rational equations involves manipulating the equation to eliminate the fractions and isolate the variable. This often requires finding a common denominator, cross-multiplication, and careful consideration of potential extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation, usually arising from the process of clearing denominators. This is why it's crucial to check solutions in the original equation.

Before diving into the specifics of our equation, let's understand the general approach to solving rational equations. First, we identify the least common denominator (LCD) of all the fractions in the equation. The LCD is the smallest expression that is divisible by each denominator. Multiplying both sides of the equation by the LCD clears the fractions, transforming the equation into a more manageable form, often a linear or quadratic equation. Once the fractions are eliminated, we can apply standard algebraic techniques to solve for the variable. Finally, we must check our solutions by substituting them back into the original equation to ensure they are valid and not extraneous.

Step-by-Step Solution

Let's break down the solution to the equation 7 = 2/(x-9) = 9/(x+12) step by step:

1. Separate the Compound Equation

The given equation is a compound equation, meaning it can be separated into two distinct equations. This allows us to handle each part individually, making the solution process more straightforward. The equation 7 = 2/(x-9) = 9/(x+12) can be split into two equations:

  • Equation 1: 7 = 2/(x-9)
  • Equation 2: 7 = 9/(x+12)

2. Solve Equation 1: 7 = 2/(x-9)

To solve the first equation, we need to isolate 'x'. This involves clearing the fraction and then using basic algebraic manipulations.

  • Multiply both sides by (x-9) to eliminate the denominator: 7(x-9) = 2
  • Distribute the 7: 7x - 63 = 2
  • Add 63 to both sides: 7x = 65
  • Divide both sides by 7: x = 65/7

3. Solve Equation 2: 7 = 9/(x+12)

Now, let's solve the second equation using a similar approach:

  • Multiply both sides by (x+12) to eliminate the denominator: 7(x+12) = 9
  • Distribute the 7: 7x + 84 = 9
  • Subtract 84 from both sides: 7x = -75
  • Divide both sides by 7: x = -75/7

4. Check for Consistency

Crucially, for the original compound equation to hold true, both derived equations must be satisfied simultaneously. This means that the value of 'x' we find should be consistent across both solutions. However, in our case, we obtained two different values for 'x':

  • x = 65/7 from Equation 1
  • x = -75/7 from Equation 2

Since the values of 'x' are different, there is no single value of 'x' that satisfies the original compound equation. Therefore, the equation has no solution.

Extraneous Solutions and Verification

As mentioned earlier, when dealing with rational equations, it's crucial to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation. This often happens when we multiply both sides of the equation by an expression containing a variable, as this can introduce solutions that make the denominator of the original equation zero, which is undefined.

In our case, we didn't find a solution that worked for both parts of the original equation. However, let's illustrate the process of checking for extraneous solutions with a hypothetical example. Suppose we found a potential solution, say x = a. To verify if it's a valid solution, we would substitute x = a into the original equation: 7 = 2/(a-9) = 9/(a+12). If both parts of the equation hold true after substitution, then x = a is a valid solution. If either part of the equation is not satisfied or if the substitution results in division by zero, then x = a is an extraneous solution and must be discarded.

Common Mistakes and How to Avoid Them

Solving rational equations can be tricky, and several common mistakes can lead to incorrect answers. Here are some pitfalls to watch out for:

  • Forgetting to distribute: When multiplying both sides of the equation by the LCD, make sure to distribute it correctly to every term. A common error is to multiply the LCD only by the fractional terms and forget to multiply it by any whole number terms.
  • Not checking for extraneous solutions: This is perhaps the most common mistake. Always substitute your solutions back into the original equation to ensure they are valid. Failing to do so can lead to accepting extraneous solutions as correct.
  • Incorrectly simplifying fractions: Before clearing denominators, simplify the fractions as much as possible. This can make the equation easier to work with and reduce the chances of making errors.
  • Making arithmetic errors: Simple arithmetic errors can easily derail the solution process. Take your time, double-check your calculations, and use a calculator if needed.

Conclusion

In conclusion, solving the equation 7 = 2/(x-9) = 9/(x+12) requires a careful and methodical approach. By separating the compound equation into two simpler equations, solving each individually, and checking for consistency, we determined that there is no solution to this particular equation. This highlights the importance of verifying solutions in rational equations, as potential solutions may not always satisfy the original equation due to the nature of rational expressions. The process reinforces key algebraic principles, including clearing denominators, isolating variables, and the critical step of checking for extraneous solutions. Understanding these principles and practicing similar problems will strengthen your ability to solve rational equations with confidence and accuracy. Remember to always double-check your work and consider potential pitfalls to ensure you arrive at the correct solution. Rational equations are a fundamental concept in algebra, and mastering them is essential for further studies in mathematics and related fields. The ability to manipulate and solve these equations is a valuable skill that will serve you well in various mathematical and real-world applications.