Question

What Is Wrong? One step in the following list contains an error. Identify it and explain what is wrong. $$ \begin{aligned} x &=2 \\ 3 x-2 x &=2 \\ 3 x &=2 x+2 \\ x^{2}+3 x &=x^{2}+2 x+2 \\ x^{2}+3 x-10 &=x^{2}+2 x-8 \\ (x-2)(x+5) &=(x-2)(x+4) \\ x+5 &=x+4 \\ 1 &=0 \end{aligned} $$

Short answer

The error is at Step 5. The subtraction resulted in an incorrect equation leading to a false statement.

Step-by-step solution

Understand the Given Problem

The problem involves an algebraic error in the following equations. The goal is to identify and explain the mistake.

Identify Each Equation

List the given equations to observe their progression:1. \( x = 2 2. 3x - 2x = 2 \)3. \( 3x = 2x + 2 \)4. \( x^2 + 3x = x^2 + 2x + 2 \)5. \( x^2 + 3x - 10 = x^2 + 2x - 8 \)6. \( (x-2)(x+5) = (x-2)(x+4) \)7. \( x+5 = x+4 \)8. \( 1 = 0 \)

Analyze Each Step for Consistency

Check every step to make sure it logically follows from the previous step.

Detail Examination of Steps

Let's start reviewing each step in detail:- From Step 1: \( x = 2 \)- Step 2: Simplifies correctly to \( 3x - 2x = 2 \), which is \( x = 2 \).- Step 3: \( 3x = 2x + 2 \) follows from the previous step by substituting \( x = 2 \).- Step 4: \( x^2 + 3x = x^2 + 2x + 2 \) by adding \( x^2 \) to both sides.- Step 5: \( x^2 + 3x - 10 = x^2 + 2x - 8 \) is incorrect.

Identify and Explain the Error

The error occurs between Step 4 and Step 5. Step 4 gave \( x^2 + 3x = x^2 + 2x + 2 \). Subtracting \( x^2 \) from both sides results in \( 3x = 2x + 2 \), not the given steps in the problem.

Correct Step 5

Correct the equation to resolve: \( 3x - 2x = 2 \) which simplifies to \( x = 2 \)

Revised Steps

To accurately solve the system without error, the equations should instead follow:1. \( x = 2 \)2. \( 3x - 2x = 2 \)3. \( x = 2 \)

Key concepts

Error Identification

Spotting errors in algebra is crucial for solving equations correctly. It helps you ensure the steps you are taking are logical and mathematically sound. In the given exercise, we encountered a critical error that disrupted the whole calculation. The error was between Step 4 and Step 5.
Let's briefly go over why this happened:
- In Step 4, we have: \( x^2 + 3x = x^2 + 2x + 2 \). This step is correct.
- But in Step 5: \( x^2 + 3x -10 = x^2 + 2x - 8 \). This transformation doesn’t follow logically from Step 4.

To identify such errors:

• Always compare each step with the previous ones to ensure consistency.
• Spot any abrupt or illogical jumps in the equation progression.

Knowing where and why the mistake occurred allows us to correct it and prevent similar errors in the future.

Equation Consistency

Maintaining consistency throughout the steps of solving an equation is essential for accurate results. Consistency ensures that each transformation is valid and that we're following algebraic rules correctly.
Here are some key points to keep in mind:

• Each new step should clearly follow from the one before.
• Any changes made to one side of the equation must be made to the other side.
• Watch for any signs that logic might be disrupted, such as contradictions or impossible results (like 1 = 0).

On analyzing the exercise:
- Step 4 gave us: \( x^2 + 3x = x^2 + 2x + 2 \)
- From here, subtracting \( x^2 \) from both sides simplifies to: \( 3x = 2x + 2 \)
-But in step 5, this wasn’t followed, leading to the error.
Properly maintaining equation consistency avoids such pitfalls.

Algebraic Simplification

Simplification is a fundamental part of solving algebraic equations. Simplifying means expressing the equation in its simplest form without changing its value. It makes the problem easier to solve and understand.

In the given exercise, proper simplification goes hand in hand with maintaining consistency. We look to transform equations in valid ways that make solving straightforward:
Consider the equations:
- Step 1: \( x = 2 \)
- Steps 2 and 3 simplify correctly to: \( 3x - 2x = 2 \) and \( 3x = 2x + 2 \)
- Step 4 further simplifies correctly, adding \( x^2 \) to get: \( x^2 + 3x = x^2 + 2x + 2 \)
- Then, step 5 should simplify by subtracting \( x^2 \) from both sides and solving for x: \( 3x = 2x +2 \).

Key tips for algebraic simplification:

• Combine like terms wherever possible.
• Make use of operations that simplify the equation logically.
• Be mindful of each operation's validity and impact on the equation's balance.

By understanding and correctly applying simplification, the solutions become clearer and more manageable.