Solving Algebraic Equations A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a code, where the unknown variable, often represented by 'x,' is the secret we aim to unlock. Samuel is on a mission to solve the equation $3(x-1)+4=2-(x+3)$, and we're here to explore the different paths he might take to reach the solution. This journey involves transforming the original equation into equivalent forms, each step bringing us closer to isolating 'x.' We'll delve into the underlying principles of equation manipulation, highlighting the importance of maintaining balance and equivalence throughout the process. Understanding these equivalent transformations is crucial not only for solving this particular equation but also for tackling a wide range of mathematical problems. So, let's embark on this algebraic adventure and dissect the potential strategies Samuel can employ.

Unraveling the Equation: Options for Samuel

When Samuel is trying to solve the equation $3(x-1)+4=2-(x+3)$, he has a few options for equivalent equations he can use. Let's break down each option and see if it aligns with the rules of algebraic manipulation. The core idea behind solving equations is to perform the same operations on both sides, ensuring the equation remains balanced. This could involve distributing terms, combining like terms, or isolating the variable. We'll examine each option through this lens, identifying the specific steps involved and verifying their validity. By understanding the rationale behind each transformation, we can appreciate the flexibility and power of algebraic techniques. This process is not just about finding the right answer; it's about developing a deep understanding of how equations work and how they can be manipulated to reveal the hidden value of the unknown.

Option A: $3x - 3 + 4 = 2 - x - 3$

This option seems to involve the distributive property, a cornerstone of algebraic manipulation. The distributive property states that a(b + c) = ab + ac. In our case, we have 3(x - 1) on the left side of the original equation. Applying the distributive property, we multiply 3 by both 'x' and '-1', resulting in 3x - 3. On the right side, we have -(x + 3), which can be treated as -1(x + 3). Distributing the -1, we get -x - 3. Thus, option A, $3x - 3 + 4 = 2 - x - 3$, correctly applies the distributive property to both sides of the original equation. This transformation maintains the equivalence of the equation, making it a valid step in solving for 'x'. Understanding the distributive property is crucial for simplifying expressions and solving equations, as it allows us to remove parentheses and combine like terms. This step lays the groundwork for further simplification and isolation of the variable.

Option B: $3x - 1 + 4 = 2 - x + 3$

This option appears to have an error in the distribution. Looking at the left side of the original equation, $3(x-1)+4$, if we distribute the 3 across the parenthesis we should get $3 * x + 3 * (-1) + 4 = 3x - 3 + 4$. Option B incorrectly states $3x - 1 + 4$, which means the 3 was not properly distributed to the -1 inside the parenthesis. On the right side of the original equation, $2-(x+3)$, distributing the negative sign across the parenthesis should give us $2 - x - 3$, but Option B incorrectly states $2 - x + 3$, which means the negative sign was not properly distributed to the +3 inside the parenthesis. Therefore, Option B does not produce an equivalent equation due to errors in the distribution process on both sides of the equation. It is crucial to pay close attention to signs and coefficients when applying the distributive property to ensure the equation remains balanced and equivalent.

Option C: $3x - 3 + 4 = 2 - x - 3$

Option C mirrors the correct application of the distributive property as seen in Option A. Let's revisit the steps: On the left side, distributing the 3 in $3(x - 1) + 4$ correctly yields $3x - 3 + 4$. On the right side, distributing the negative sign in $2 - (x + 3)$ results in $2 - x - 3$. Thus, the equation $3x - 3 + 4 = 2 - x - 3$ accurately reflects the application of the distributive property on both sides of the original equation. This makes Option C a valid equivalent equation that Samuel could use. The ability to correctly apply the distributive property is a fundamental skill in algebra, allowing us to simplify expressions and move closer to solving for the unknown variable. Recognizing and avoiding errors in distribution, as we saw in Option B, is equally important.

Option D: $3x + 1 = -x - 1$

This option represents a further step in simplifying the equation. To verify its validity, let's start with the equivalent equation we derived in Options A and C: $3x - 3 + 4 = 2 - x - 3$. The next logical step is to combine the constant terms on each side. On the left side, -3 + 4 simplifies to +1. On the right side, 2 - 3 simplifies to -1. This gives us the equation $3x + 1 = -x - 1$, which is exactly what Option D presents. Therefore, Option D is indeed an equivalent equation that Samuel could use. This step demonstrates the importance of combining like terms to simplify equations and make them easier to solve. By reducing the number of terms on each side, we bring the equation closer to a form where the variable can be isolated.

The Significance of Equivalent Equations

The concept of equivalent equations is central to solving algebraic problems. Equivalent equations are different forms of the same equation that have the same solution set. This means that any value of 'x' that satisfies one equation will also satisfy its equivalent forms. The power of using equivalent equations lies in the ability to manipulate the original equation into a simpler form that is easier to solve. This manipulation involves applying various algebraic operations, such as the distributive property, combining like terms, adding or subtracting the same value from both sides, and multiplying or dividing both sides by the same non-zero value. Each of these operations transforms the equation while preserving its fundamental balance and solution set. Understanding and utilizing equivalent equations is a crucial skill for anyone venturing into the world of algebra and beyond. It allows us to approach complex problems with a systematic and logical approach, breaking them down into manageable steps.

Conclusion

In conclusion, Samuel has multiple paths to solve the equation $3(x-1)+4=2-(x+3)$. Options A, C, and D represent valid equivalent equations that Samuel might use in his solving process. Option B, however, contains errors in the distribution and does not represent an equivalent equation. By carefully applying algebraic principles and verifying each step, Samuel can confidently navigate the equation-solving journey and arrive at the correct solution. The key takeaway is that solving equations is not just about finding the answer; it's about understanding the underlying principles and developing the ability to manipulate equations while preserving their equivalence. This skill is invaluable not only in mathematics but also in various fields that require problem-solving and analytical thinking.