Problem 42 - Algebra 1: Concepts and Skills

Question

Find and correct the error. \(\frac{1}{4}(x-2)=8\) \(x+2=2\) \(x=4\)

Step-by-Step Solution

Answer

The error was in the second operation \(x+2=2\). The correct operation should be \(x-2=32\). Thus, the correct solution is \(x=34\).

Step 1: Identify the first correct operation

The first operation, i.e., \(\frac{1}{4}(x-2)=8\), is correct. Therefore, it is necessary to isolate the variable on one side of the equation. Multiply both sides by 4 to do this.

Step 2: Correct the error in the second operation

From the product of 8 and 4, the result should be 32, not 2. Hence the second operation should be \(x-2=32\) instead of \(x+2=2\).

Step 3: Solve for x

Finally, to solve for the variable x, add 2 on both sides of the equation. This leads to the correct value of x to be \(x=34\).

Key Concepts

Algebra Fundamentals

Algebra is like the puzzle-solver’s toolkit. It’s all about finding the unknowns represented by variables, usually denoted as letters like \( x \), and understanding the relationships between them through equations. Equations express two things as equal, containing both known values (constants) and unknowns (variables). For instance, in the exercise \( \frac{1}{4}(x-2)=8 \), the goal is to 'solve for \( x \)'. This means figuring out which value of \( x \) makes the equation true.

To do this, we follow systematic steps, manipulating the equation to isolate the variable on one side. This process often involves basic operations like addition or subtraction to both sides of an equation, so the equation stays balanced, just like a see-saw. Algebra is foundational and appears everywhere in mathematics, forming the base for more complex concepts like calculus and statistics.

Spotting and Correcting Errors

Mistakes in solving equations are normal, and learning to spot and correct them is crucial. They often happen during one of the mathematical operations or when translating steps mentally into paper. Here’s how you can make sure you’re on the right track.

• Double-check each arithmetic operation you perform. Small errors in calculations can lead to big mistakes in results.
• Verify each rearranged equation step by reversing operations, which can help confirm the step's validity. For example, if you multiplied both sides by 4, check the multiplication carefully, as done in the exercise where applying 4 transforms \( x-2=8 \) to \( x-2=32 \). But the misstep occurred as \( x+2=2 \) instead of correctly formulating \( x-2=32 \).
• Review your operations logically. After each step, see if the new equation makes sense and aligns with your expected outcome.

Correctly solving a problem not only involves reaching the solution but making sure everything in between is valid and well-executed. By regularly practicing error identification and correction, you'll become proficient in evaluating your mathematical work.

Step by Step Solutions

Breaking down a problem step by step makes complex problems more digestible. Let’s take a look at the process for solving equations like our example in the original exercise.

• Step 1: Understand the Equation. Begin by looking at the equation carefully, identifying your constants and variables. Here, \( \frac{1}{4}(x-2)=8 \), the first step involves distributing or dealing with fractions effectively.
• Step 2: Isolate the Variable. We multiply or divide to eliminate fractions. For this problem, multiplying each side by 4 eliminates the fraction, leading to an easier-to-solve equation.
• Step 3: Correct Each Step. Instead of proceeding straight to adding or subtracting indiscriminately, confirm that you correctly performed the prior multiplication before simplifying further. As seen, the misstep \( x+2=2 \) was identified and corrected to \( x-2=32 \) followed by adding 2 to achieve \( x=34 \).

Working through problems one step at a time not only helps prevent errors but builds your ability to approach even more challenging problems. Ensure you reflect on why each step is necessary and how it serves to simplify or solve the equation effectively.