Solving $-7(x-2)+4=6(x+5)$ For X Algebraically

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In the realm of algebra, solving equations is a fundamental skill. The ability to manipulate equations and isolate variables is crucial for various mathematical and scientific applications. In this comprehensive guide, we will delve into the process of solving the equation $-7(x-2)+4=6(x+5)$ for $x$ algebraically. Our aim is to provide a clear and step-by-step explanation, ensuring that you grasp the underlying concepts and can confidently tackle similar problems.

Understanding the Equation

The equation we are tasked with solving is $-7(x-2)+4=6(x+5)$. This is a linear equation, which means that the highest power of the variable $x$ is 1. Linear equations are relatively straightforward to solve, and the general strategy involves isolating the variable on one side of the equation. To achieve this, we will employ a series of algebraic manipulations, ensuring that we maintain the equality throughout the process.

Step 1: Distribute

The first step in solving the equation is to eliminate the parentheses by distributing the coefficients. Distributing involves multiplying the term outside the parentheses by each term inside the parentheses. Let's apply this to our equation:

$-7(x-2)+4=6(x+5)$

Distribute the -7 on the left side:

$-7 * x + (-7) * (-2) + 4 = 6(x+5)$

$-7x + 14 + 4 = 6(x+5)$

Now, distribute the 6 on the right side:

$-7x + 14 + 4 = 6 * x + 6 * 5$

$-7x + 14 + 4 = 6x + 30$

Step 2: Simplify

Next, we simplify both sides of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, we can combine the constant terms on the left side:

$-7x + 14 + 4 = 6x + 30$

$-7x + 18 = 6x + 30$

Step 3: Isolate the Variable

Our goal is to isolate the variable $x$ on one side of the equation. To do this, we need to move all the terms containing $x$ to one side and all the constant terms to the other side. Let's start by moving the $-7x$ term from the left side to the right side. We can do this by adding $7x$ to both sides of the equation:

$-7x + 18 + 7x = 6x + 30 + 7x$

$18 = 13x + 30$

Now, let's move the constant term 30 from the right side to the left side. We can do this by subtracting 30 from both sides of the equation:

$18 - 30 = 13x + 30 - 30$

$-12 = 13x$

Step 4: Solve for x

Finally, to solve for $x$, we need to isolate it completely. We can do this by dividing both sides of the equation by the coefficient of $x$, which is 13:

$-12 / 13 = 13x / 13$

$x = -12/13$

Therefore, the solution to the equation $-7(x-2)+4=6(x+5)$ is $x = -12/13$.

Expressing the Answer in Reduced Fractional Form

The problem statement specifically requests that if the answer is a fraction, it should be written in reduced, fractional form. Our answer, $-12/13$, is already in reduced form because the numerator (-12) and the denominator (13) have no common factors other than 1. This means the fraction cannot be simplified further.

Avoiding Decimal Conversion

The instructions also explicitly state that the answer should not be converted to a decimal. Keeping the answer in fractional form provides an exact representation of the solution. Converting to a decimal might introduce rounding errors, which can be problematic in certain contexts.

Key Concepts and Techniques

Let's recap the key concepts and techniques we employed to solve the equation:

• Distribution: This involves multiplying a term outside parentheses by each term inside the parentheses.
• Simplifying: This involves combining like terms on both sides of the equation.
• Isolating the Variable: This involves moving all terms containing the variable to one side of the equation and all constant terms to the other side.
• Solving for the Variable: This involves dividing both sides of the equation by the coefficient of the variable.
• Reduced Fractional Form: This means expressing a fraction in its simplest form, where the numerator and denominator have no common factors other than 1.

Common Mistakes to Avoid

When solving algebraic equations, it's crucial to be mindful of common mistakes that can lead to incorrect answers. Here are a few to watch out for:

• Incorrect Distribution: Make sure to distribute the coefficient to every term inside the parentheses. Pay close attention to signs (positive and negative).
• Combining Unlike Terms: Only combine terms that have the same variable raised to the same power. For example, you cannot combine $-7x$ and 18.
• Incorrectly Applying Operations: When moving terms from one side of the equation to the other, remember to perform the opposite operation. For example, to move a term that is being added, you need to subtract it from both sides.
• Forgetting to Simplify: Always simplify both sides of the equation as much as possible before proceeding with other steps.
• Rounding Errors: Avoid converting fractions to decimals unless explicitly instructed to do so. Decimal approximations can introduce rounding errors.

Extending Your Understanding

The techniques we've discussed for solving this equation can be applied to a wide range of linear equations. The key is to follow the steps systematically and pay attention to detail. To further enhance your understanding, consider practicing with various examples and gradually increasing the complexity of the equations.

Real-World Applications

Solving algebraic equations is not just a theoretical exercise; it has numerous real-world applications. Linear equations, in particular, are used to model various phenomena in fields such as:

• Physics: Calculating motion, forces, and energy.
• Engineering: Designing structures, circuits, and systems.
• Economics: Analyzing supply and demand, predicting market trends.
• Finance: Calculating interest rates, loan payments, and investment returns.
• Computer Science: Developing algorithms, solving optimization problems.

By mastering the art of solving algebraic equations, you equip yourself with a powerful tool that can be applied to a vast array of problems.

Conclusion

In this guide, we have meticulously walked through the process of solving the equation $-7(x-2)+4=6(x+5)$ for $x$ algebraically. We've emphasized the importance of each step, from distribution and simplification to isolating the variable and expressing the answer in reduced fractional form. By understanding the underlying concepts and practicing regularly, you can develop the skills necessary to confidently solve a wide range of algebraic equations. Remember to avoid common mistakes, focus on accuracy, and appreciate the real-world applications of this fundamental mathematical skill.

• Distribution is crucial for eliminating parentheses.
• Simplifying both sides of the equation makes it easier to work with.
• Isolating the variable is the key to finding its value.
• Reduced fractional form provides an exact representation of the solution.
• Practice is essential for mastering algebraic equation solving.

By following these guidelines, you'll be well-equipped to tackle algebraic equations with confidence and precision. Remember, algebra is a foundational skill that opens doors to a world of mathematical and scientific possibilities.

Answer:

$x = -12/13$