Solving Equations: Liliana's Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of solving equations, a fundamental concept in mathematics that opens doors to understanding various real-world problems. Today, we'll analyze Liliana's attempt to solve a simple equation and pinpoint the correction she needs to make. This journey will not only enhance your equation-solving skills but also sharpen your critical thinking abilities. So, buckle up, grab your pencils, and let's get started!

The Equation and Liliana's Attempt

Let's consider the equation: {rac{1}{3}(x+18) = 7}. Liliana's solution steps are as follows:

Our task is to identify where Liliana might have stumbled. Let's meticulously examine each step to find any errors in her logic or calculations. Remember, solving equations is all about precision and following the correct mathematical rules. We will be using our knowledge of the order of operations and the properties of equality to carefully evaluate Liliana's work and find any potential mistakes. Analyzing the steps one by one is crucial, as each step builds upon the previous one; any error in a preceding step will inevitably lead to an incorrect final answer. Understanding how equations work is a key concept in Algebra and is fundamental to further studies in mathematics, so let's check it out!

Step-by-Step Breakdown

• Step 1: ${\frac{1}{3}(x+18) = 7}$ - This is the original equation, and it's perfectly set up. No issues here. It clearly defines the problem we are trying to solve. From this step we will see the steps of working that will lead us to the correct solution.
• Step 2: ${\frac{1}{3}x + 6 = 7}$ - This is where the distributive property is applied. Multiplying ${\frac{1}{3}}$ by both ${x}$ and ${18}$ should result in ${\frac{1}{3}x + 6}$. This step is done correctly. The distributive property has been correctly applied, and we can move to the next step.
• Step 3: ${\frac{1}{3}x + 6 - 6 = 7 + 6}$ - To isolate the term with ${x}$, Liliana correctly subtracts 6 from the left side. However, to maintain the equality, she should also subtract 6 from the right side, not add it. This is a crucial mistake, as it violates the properties of equality. When you perform an operation on one side of the equation, you must perform the same operation on the other side to keep the equation balanced.
• Step 4: ${\frac{1}{3}x = 1}$ - Based on the incorrect previous step, the equation simplifies to ${\frac{1}{3}x = 1}$. Due to the error in the previous step, this step is also incorrect.
• Step 5: ${x = 3}$ - Liliana multiplies both sides by 3 to solve for ${x}$, which is the correct operation to isolate ${x}$. However, because the equation ${\frac{1}{3}x = 1}$ was incorrect, her final solution is also incorrect. The value of x does not match the original equation.

Identifying the Error

Liliana's error lies in Step 3. Instead of subtracting 6 from both sides of the equation, she added 6 to the right side. This mistake disrupted the balance of the equation and led to an incorrect solution. The fundamental rule of solving equations is to perform the same operation on both sides to maintain equality. By not following this rule, Liliana introduced an error that propagated through the remaining steps, leading to an incorrect result.

Why This Matters

Understanding the importance of this rule is crucial for success in algebra and beyond. It underpins all equation-solving techniques and allows us to manipulate equations without changing their underlying meaning. If you break this rule, any subsequent steps will be useless, as they will build upon an erroneous foundation. The goal is to always manipulate equations in such a way that the solution remains true for the original equation. In real life, being able to accurately solve equations is crucial in various fields, such as engineering, finance, and computer science. The basic principles of algebra can be applied to real-world problems. For example, solving linear equations can help determine the rate of a project or in understanding financial models. By mastering these principles, students can establish a foundation for more advanced math concepts and gain valuable problem-solving skills.

Correcting the Solution

Let's go through the steps again, this time correctly:

1. ${\frac{1}{3}(x+18) = 7}$
2. ${\frac{1}{3}x + 6 = 7}$
3. ${\frac{1}{3}x + 6 - 6 = 7 - 6}$ (Subtract 6 from both sides)
4. ${\frac{1}{3}x = 1}$
5. ${3 * \frac{1}{3}x = 1 * 3}$ (Multiply both sides by 3)
6. ${x = 3}$

In step 3, we subtract 6 from both sides, which isolates ${\frac{1}{3}x}$. Then, in step 5, we multiply both sides by 3 to solve for ${x}$. So, the answer is x = 3, which is the correct solution. To make sure that the answer is right, we can test it by inserting the number in the original equation, which will make the equation true. It is always a good idea to test your solution, especially when learning. This helps build confidence and provides immediate confirmation of the answer's accuracy.

Conclusion: Mastering Equation Solving

So, guys, what's the takeaway? Liliana's mistake in Step 3 highlights the crucial need to maintain balance in an equation. Always remember to perform the same operation on both sides to keep the equality intact. Mastering equation solving is a building block in mathematics. By understanding each step and the rules that govern them, we can solve various problems with confidence. Practice is key! The more you work through equations, the more familiar you will become with the properties and techniques. Don't be afraid to make mistakes; they are part of the learning process. Each time you solve an equation, you're sharpening your problem-solving skills. Keep practicing, and you'll find yourself solving complex equations like a pro in no time! Keep up the good work and keep practicing!