Solving $6x - 2 = -4x + 2$ Valid Approaches By Spencer And Jeremiah

Solving linear equations is a fundamental skill in algebra, and often, there are multiple paths to arrive at the correct solution. In this article, we'll dissect the equation $6x - 2 = -4x + 2$ and evaluate two different approaches suggested by Spencer and Jeremiah. Spencer proposes adding $4x$ to both sides as the initial step, while Jeremiah suggests subtracting $6x$ from both sides. To determine who is correct, we'll walk through each method, highlighting the underlying algebraic principles and demonstrating why both approaches are valid.

Understanding the Equation $6x - 2 = -4x + 2$

Before diving into the solution methods, it's crucial to understand the structure of the equation $6x - 2 = -4x + 2$. This is a linear equation in one variable, $x$. The goal is to isolate $x$ on one side of the equation to find its value. The equation states that the expression on the left-hand side ($6x - 2$) is equal to the expression on the right-hand side ($-4x + 2$). To solve for $x$, we need to manipulate the equation while maintaining this equality. This is achieved by performing the same operations on both sides of the equation. The fundamental principle behind solving equations is to perform inverse operations to isolate the variable. We can add, subtract, multiply, or divide both sides of the equation by the same value without changing the solution. Let's explore how Spencer's and Jeremiah's approaches align with this principle.

Spencer's Approach: Adding $4x$ to Both Sides

Spencer suggests adding $4x$ to both sides of the equation as the first step. This approach is perfectly valid and demonstrates a strong understanding of algebraic manipulation. Let's examine why this works and how it progresses the solution.

The initial equation is:

$6x - 2 = -4x + 2$

Adding $4x$ to both sides, we get:

$6x - 2 + 4x = -4x + 2 + 4x$

Now, simplify both sides by combining like terms. On the left-hand side, $6x$ and $4x$ combine to give $10x$. On the right-hand side, $-4x$ and $4x$ cancel each other out, leaving just 2. The equation now looks like this:

$10x - 2 = 2$

By adding $4x$ to both sides, Spencer effectively eliminated the $x$ term from the right-hand side, which is a significant step toward isolating $x$. The next step would be to isolate the term with $x$ by adding 2 to both sides:

$10x - 2 + 2 = 2 + 2$

This simplifies to:

$10x = 4$

Finally, to solve for $x$, divide both sides by 10:

$10x / 10 = 4 / 10$

Which gives us:

$x = 2 / 5$ or $x = 0.4$

Spencer's method is a direct and efficient way to solve the equation. By adding $4x$ to both sides, he strategically simplified the equation, bringing it closer to the solution. This approach highlights the importance of recognizing opportunities to eliminate terms and streamline the equation-solving process.

Jeremiah's Approach: Subtracting $6x$ from Both Sides

Jeremiah proposes subtracting $6x$ from both sides as the initial step. This approach is also correct and offers a different perspective on how to solve the equation. Let's analyze Jeremiah's method and see how it leads to the same solution.

Starting with the original equation:

$6x - 2 = -4x + 2$

Subtracting $6x$ from both sides gives:

$6x - 2 - 6x = -4x + 2 - 6x$

Simplify both sides by combining like terms. On the left-hand side, $6x$ and $-6x$ cancel each other out, leaving $-2$. On the right-hand side, $-4x$ and $-6x$ combine to give $-10x$. The equation now looks like this:

$-2 = -10x + 2$

By subtracting $6x$ from both sides, Jeremiah eliminated the $x$ term from the left-hand side, which is another valid strategy. The next step would be to isolate the term with $x$ by subtracting 2 from both sides:

$-2 - 2 = -10x + 2 - 2$

This simplifies to:

$-4 = -10x$

To solve for $x$, divide both sides by -10:

$-4 / -10 = -10x / -10$

Which gives us:

$x = 2 / 5$ or $x = 0.4$

Jeremiah's method demonstrates that there is often more than one way to solve an equation. By subtracting $6x$ from both sides, he took a different route but arrived at the same correct answer. This approach emphasizes the flexibility in algebraic problem-solving and the importance of understanding the properties of equality.

Comparing the Two Approaches

Both Spencer's and Jeremiah's approaches are valid and lead to the same solution, $x = 2 / 5$ or $x = 0.4$. The key difference lies in the initial manipulation of the equation. Spencer chose to add $4x$ to both sides, which eliminated the negative $x$ term on the right-hand side. Jeremiah chose to subtract $6x$ from both sides, which eliminated the $x$ term on the left-hand side.

The choice between these approaches often comes down to personal preference or a sense of which method might be more efficient in a particular situation. Some students might prefer to avoid negative coefficients, which could make Spencer's approach more appealing. Others might find Jeremiah's method equally straightforward. The most important takeaway is that understanding the underlying algebraic principles allows for flexibility in problem-solving.

Conclusion: Both Spencer and Jeremiah are Correct

In conclusion, both Spencer and Jeremiah are correct in their proposed first steps for solving the equation $6x - 2 = -4x + 2$. Spencer's approach of adding $4x$ to both sides and Jeremiah's approach of subtracting $6x$ from both sides are both valid algebraic manipulations that maintain the equality of the equation. This example illustrates a fundamental concept in algebra: there can be multiple correct ways to solve an equation. The most effective approach often depends on individual preference and the specific characteristics of the equation. By understanding the properties of equality and the principles of inverse operations, students can confidently navigate different solution paths and arrive at the correct answer. Mastering these skills is crucial for success in algebra and more advanced mathematical topics.

This problem emphasizes the importance of understanding the properties of equality and the flexibility in solving algebraic equations. It's not about finding the right way, but about understanding a right way and being able to justify each step. This kind of conceptual understanding is key to building a strong foundation in mathematics.