Solve The Equation $-2(-8x-7) = -3(-6-x) - 8x$ Algebraically. Answer As A Reduced Proper Or Improper Fraction Only.$x =$ $\square$

Introduction

In this article, we will delve into the world of algebra and solve the given equation $-2(-8x-7) = -3(-6-x) - 8x$ step by step. We will use algebraic techniques to simplify the equation and find the value of $x$. Our goal is to provide a clear and concise solution to the equation, and we will use proper mathematical notation and terminology throughout the article.

Step 1: Distribute the Negative Signs

The first step in solving the equation is to distribute the negative signs on the left-hand side of the equation. This will allow us to simplify the equation and make it easier to work with.

$$-2(-8x-7) = -3(-6-x) - 8x$$

Distributing the negative signs on the left-hand side, we get:

$$2(8x+7) = -3(-6-x) - 8x$$

Using the distributive property, we can simplify the left-hand side of the equation:

$$16x+14 = -3(-6-x) - 8x$$

Step 2: Simplify the Right-Hand Side

Next, we need to simplify the right-hand side of the equation. We can start by distributing the negative sign on the right-hand side:

$$16x+14 = 3(6+x) - 8x$$

Using the distributive property, we can simplify the right-hand side of the equation:

$$16x+14 = 18+3x-8x$$

Combining like terms on the right-hand side, we get:

$$16x+14 = 18-5x$$

Step 3: Move All Terms to One Side

Now that we have simplified both sides of the equation, we can move all the terms to one side of the equation. We will move all the terms to the left-hand side of the equation:

$$16x+14 = 18-5x$$

Subtracting $18$ from both sides of the equation, we get:

$$16x-4 = -5x$$

Adding $5x$ to both sides of the equation, we get:

$$21x-4 = 0$$

Step 4: Add 4 to Both Sides

Next, we need to add $4$ to both sides of the equation to isolate the term with $x$:

$$21x-4+4 = 0+4$$

Simplifying the left-hand side of the equation, we get:

$$21x = 4$$

Step 5: Divide Both Sides by 21

Finally, we need to divide both sides of the equation by $21$ to solve for $x$:

$$\frac{21x}{21} = \frac{4}{21}$$

Simplifying the left-hand side of the equation, we get:

$$x = \frac{4}{21}$$

Conclusion

In this article, we solved the equation $-2(-8x-7) = -3(-6-x) - 8x$ step by step using algebraic techniques. We distributed the negative signs, simplified the right-hand side, moved all the terms to one side, added $4$ to both sides, and finally divided both sides by $21$ to solve for $x$. Our final answer is $x = \frac{4}{21}$.

Final Answer

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Q&A: Frequently Asked Questions

Q: What is the first step in solving the equation $-2(-8x-7) = -3(-6-x) - 8x$? A: The first step in solving the equation is to distribute the negative signs on the left-hand side of the equation.

Q: How do I distribute the negative signs on the left-hand side of the equation? A: To distribute the negative signs, you need to multiply each term inside the parentheses by the negative sign. In this case, you would multiply $-8x$ and $-7$ by $-2$.

Q: What is the result of distributing the negative signs on the left-hand side of the equation? A: The result of distributing the negative signs on the left-hand side of the equation is $2(8x+7)$.

Q: How do I simplify the left-hand side of the equation? A: To simplify the left-hand side of the equation, you need to use the distributive property to multiply $2$ by each term inside the parentheses. In this case, you would multiply $2$ by $8x$ and $2$ by $7$.

Q: What is the result of simplifying the left-hand side of the equation? A: The result of simplifying the left-hand side of the equation is $16x+14$.

Q: How do I simplify the right-hand side of the equation? A: To simplify the right-hand side of the equation, you need to distribute the negative sign on the right-hand side and then combine like terms.

Q: What is the result of simplifying the right-hand side of the equation? A: The result of simplifying the right-hand side of the equation is $18-5x$.

Q: How do I move all the terms to one side of the equation? A: To move all the terms to one side of the equation, you need to add or subtract the same value from both sides of the equation. In this case, you would add $5x$ to both sides of the equation.

Q: What is the result of moving all the terms to one side of the equation? A: The result of moving all the terms to one side of the equation is $21x-4 = 0$.

Q: How do I add 4 to both sides of the equation? A: To add 4 to both sides of the equation, you need to add 4 to the left-hand side of the equation and add 4 to the right-hand side of the equation.

Q: What is the result of adding 4 to both sides of the equation? A: The result of adding 4 to both sides of the equation is $21x = 4$.

Q: How do I divide both sides of the equation by 21? A: To divide both sides of the equation by 21, you need to divide the left-hand side of the equation by 21 and divide the right-hand side of the equation by 21.

Q: What is the result of dividing both sides of the equation by 21? A: The result of dividing both sides of the equation by 21 is $x = \frac{4}{21}$.

Conclusion

In this article, we provided a step-by-step guide to solving the equation $-2(-8x-7) = -3(-6-x) - 8x$. We also answered frequently asked questions about the equation and the steps involved in solving it. Our final answer is $x = \frac{4}{21}$.

Final Answer

The final answer is $\boxed{\frac{4}{21}}$.