Answered

Luke solved a rational equation as shown, but his work contains an error. Identify the step in which Luke made an error.



\multicolumn{2}{|c|}{$\frac{1}{x}+\frac{1}{x-4}=\frac{x-3}{x-4}$}

Step 1:		$\frac{1}{x}[x(x-4)]+\frac{1}{x-4}[x(x-4)]=\frac{x-3}{x-4}[x(x-4)]$

Step 2:		$\frac{x(x-4)}{x}+\frac{x(x-4)}{x-4}=\frac{x(x-3)(x-4)}{x-4}$

Step 3:		$x-4+x-4=x(x-4)$

Step 4:		$2 x-8=x^2-4 x$

Step 5:		$0=x^2-6 x+8$

Step 6:		$0=(x-2)(x-4)$

Step 7:		$\begin{array}{c}
0=x-2 \text { or } 0=x-4

x=2, x=4
\end{array}$

Verified Answer :

  • The original rational equation is \(\frac{1}{x} + \frac{1}{x-4} = \frac{x-3}{x-4}\).
  • Multiplying by the LCD \(x(x-4)\) correctly leads to \(\frac{x(x-4)}{x} + \frac{x(x-4)}{x-4} = \frac{x(x-3)(x-4)}{x-4}\).
  • Simplifying these terms results in \(x-4 + x = x(x-3)\).
  • Luke's Step 3 erroneously represents this as \(x-4 + x-4 = x(x-4)\), identifying it as the first error.
  • The error is in \(\boxed{Step 3}\).

Explanation

  1. Problem Analysis

The objective is to identify the step in which Luke made an error while solving the rational equation \(\frac{1}{x} + \frac{1}{x-4} = \frac{x-3}{x-4}\). First, we note the original equation and the steps taken. The least common denominator (LCD) for the terms in the equation is \(x(x-4)\). Multiplying each term by \(x(x-4)\) as shown in Step 1 is correct: \(\frac{x(x-4)}{x} + \frac{x(x-4)}{x-4} = \frac{x(x-3)(x-4)}{x-4}\). This step is verified.

  1. Identifying the Error

Next, let us perform the algebraic simplifications starting from Step 2.

  • The first term: \(\frac{x(x-4)}{x} = x-4\)
  • The second term: \(\frac{x(x-4)}{x-4} = x\)
  • The right side: \(\frac{x(x-3)(x-4)}{x-4} = x(x-3)\)

Comparing this to the equation in Step 3, we have:

  • Correct form: \(x-4 + x = x(x-3)\)
  • Luke's form: \(x-4 + x-4 = x(x-4)\)

It is evident that Luke incorrectly simplified the terms. In the second term of the left side, he wrote \(x-4\) instead of \(x\), and on the right side, he wrote \(x(x-4)\) instead of \(x(x-3)\). Thus, the error occurs in Step 3.

  1. Conclusion

Since the error occurred in Step 3, the subsequent steps (Step 4, Step 5, etc.) are based on an incorrect equation. Even though the algebraic operations in the later steps follow correctly from the faulty Step 3, the entire solution deviates from the original problem at this point. Therefore, Step 3 is indeed the step containing the first error.

Examples

Understanding where a mistake occurs in a multi-step algebraic derivation is crucial for learning. In real-world applications like engineering or finance, rational equations are often used to model rates, such as travel speed or work output. For example, if two machines work together at different rates to complete a task, the time required can be modeled by a rational equation. Identifying a calculation error in such models prevents incorrect estimations or failed processes, ensuring that resources and time are managed effectively.

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