Solve The Equation: (6-8x)/10 = (8-6x)/8 - (10-4x)/4. What Is The Correct Answer From The Options: 1) 2, 2) 42, 3) 84, 4) None Of The Above?

Jun 18, 2025 by ADMIN 141 views

Solving the Equation (6-8x)/10 = (8-6x)/8 - (10-4x)/4: A Step-by-Step Guide

This article provides a comprehensive guide to solving the equation (6-8x)/10 = (8-6x)/8 - (10-4x)/4. We will break down each step in detail, ensuring a clear understanding of the solution process. This is a common type of problem encountered in algebra, and mastering it will enhance your problem-solving skills.

Understanding the Problem

The given equation is a linear equation in one variable, x. Our goal is to isolate x on one side of the equation to find its value. The equation involves fractions, so we will need to eliminate these fractions to simplify the equation and make it easier to solve. The steps involved typically include finding a common denominator, clearing the fractions, simplifying the equation, and isolating the variable.

Step 1: Finding the Least Common Multiple (LCM)

The first crucial step in solving this equation is to eliminate the fractions. To do this, we need to find the least common multiple (LCM) of the denominators, which are 10, 8, and 4. The LCM is the smallest number that is a multiple of all the denominators. Finding the LCM is vital because it allows us to multiply both sides of the equation by a single value, effectively clearing the fractions without changing the equation's balance. Let's break down how to find the LCM:

Prime Factorization: We start by finding the prime factorization of each denominator:
- 10 = 2 × 5
- 8 = 2 × 2 × 2 = 2³
- 4 = 2 × 2 = 2²
Identify Highest Powers: Next, we identify the highest power of each prime factor that appears in any of the factorizations. In this case, we have the prime factors 2 and 5. The highest power of 2 is 2³ (from the factorization of 8), and the highest power of 5 is 5¹ (from the factorization of 10).
Multiply the Highest Powers: Finally, we multiply these highest powers together to get the LCM:
- LCM(10, 8, 4) = 2³ × 5 = 8 × 5 = 40

Therefore, the least common multiple of 10, 8, and 4 is 40. This means that 40 is the smallest number that all three denominators (10, 8, and 4) divide into evenly. This is a critical piece of information for the next step, where we will multiply both sides of the equation by 40 to clear the fractions.

By using the LCM, we ensure that each fraction in the equation will be multiplied by a value that cancels out its denominator, resulting in an equation without fractions. This significantly simplifies the problem and makes it easier to solve for x. Understanding the process of finding the LCM is a fundamental skill in algebra and is essential for solving equations involving fractions. The prime factorization method is a reliable way to find the LCM, especially when dealing with larger numbers.

Step 2: Clearing the Fractions

Now that we have determined the LCM to be 40, we can proceed to clear the fractions from the equation. This step involves multiplying both sides of the equation by the LCM. Multiplying both sides by the same number maintains the equality, while clearing the fractions simplifies the equation significantly.

The original equation is:

(6 - 8x) / 10 = (8 - 6x) / 8 - (10 - 4x) / 4

To clear the fractions, we multiply both sides of the equation by 40:

40 * [(6 - 8x) / 10] = 40 * [(8 - 6x) / 8 - (10 - 4x) / 4]

Now, we distribute the 40 on both sides of the equation. On the left side, 40 is multiplied by the fraction (6 - 8x) / 10. On the right side, we need to distribute 40 to both terms within the brackets:

40 * (6 - 8x) / 10 = 40 * (8 - 6x) / 8 - 40 * (10 - 4x) / 4

Next, we simplify each term by canceling out the common factors. For the first term, 40 divided by 10 is 4. For the second term, 40 divided by 8 is 5. For the third term, 40 divided by 4 is 10. This gives us:

4 * (6 - 8x) = 5 * (8 - 6x) - 10 * (10 - 4x)

This simplified equation no longer contains any fractions, making it much easier to work with. The process of clearing fractions is a fundamental technique in algebra, and it is often the first step in solving equations that involve fractions. By multiplying both sides of the equation by the LCM, we ensure that each denominator cancels out, leaving us with an equation that only involves integers and variables.

This step is crucial because it transforms a complex equation with fractions into a simpler, more manageable form. The distribution and simplification must be performed carefully to avoid errors. Once the fractions are cleared, the next step is to further simplify the equation by expanding the brackets and combining like terms.

Step 3: Expanding the Brackets and Simplifying

With the fractions cleared, the equation now looks like this:

4 * (6 - 8x) = 5 * (8 - 6x) - 10 * (10 - 4x)

The next step is to expand the brackets by distributing the numbers outside the parentheses to the terms inside. This will remove the parentheses and allow us to combine like terms. Expanding the brackets involves applying the distributive property, which states that a(b + c) = ab + ac.

First, we expand the left side of the equation:

4 * (6 - 8x) = 4 * 6 - 4 * 8x = 24 - 32x

Next, we expand the right side of the equation. Remember to distribute both the 5 and the -10:

5 * (8 - 6x) = 5 * 8 - 5 * 6x = 40 - 30x -10 * (10 - 4x) = -10 * 10 + (-10) * (-4x) = -100 + 40x

Now, we combine these results to get the expanded right side:

40 - 30x - 100 + 40x

So the equation now looks like:

24 - 32x = 40 - 30x - 100 + 40x

Next, we simplify the right side by combining like terms. We have two constant terms (40 and -100) and two x terms (-30x and 40x). Combining like terms means adding or subtracting terms that have the same variable and exponent.

40 - 100 = -60 -30x + 40x = 10x

So the simplified right side is:

-60 + 10x

Now the equation becomes:

24 - 32x = -60 + 10x

This simplified equation is much easier to solve. We have successfully expanded the brackets and combined like terms, which is a crucial step in solving algebraic equations. The next step involves isolating the variable x on one side of the equation.

Step 4: Isolating the Variable x

After expanding the brackets and simplifying, our equation is:

24 - 32x = -60 + 10x

To solve for x, we need to isolate it on one side of the equation. This involves moving all terms containing x to one side and all constant terms to the other side. The key principle here is to perform the same operation on both sides of the equation to maintain the equality. Isolating the variable is a fundamental technique in algebra, and it's essential for finding the value of the unknown.

First, let's move all the x terms to one side. We can do this by adding 32x to both sides of the equation. Adding 32x to both sides will eliminate the -32x term on the left side:

24 - 32x + 32x = -60 + 10x + 32x

This simplifies to:

24 = -60 + 42x

Now, we need to move the constant term (-60) to the left side of the equation. We can do this by adding 60 to both sides:

24 + 60 = -60 + 42x + 60

This simplifies to:

84 = 42x

Now we have all the x terms on one side and all the constant terms on the other side. The equation is now in a form where we can easily solve for x. The next step is to divide both sides by the coefficient of x.

The process of isolating the variable often involves multiple steps and requires careful attention to signs and operations. Each step is crucial in simplifying the equation and bringing us closer to the solution. By performing the same operations on both sides of the equation, we ensure that the equality is maintained, and we are ultimately able to find the value of the unknown variable.

Step 5: Solving for x

Our equation is now:

84 = 42x

To solve for x, we need to divide both sides of the equation by the coefficient of x, which is 42. This will isolate x on one side and give us its value. Solving for x is the final step in our process, where we determine the numerical value that satisfies the equation.

Divide both sides by 42:

84 / 42 = 42x / 42

This simplifies to:

2 = x

Therefore, the solution to the equation is x = 2. This means that when x is equal to 2, the original equation is true. We have successfully solved for x by isolating it and performing the necessary operations.

The solution x = 2 can be verified by substituting this value back into the original equation to check if both sides are equal. This is a good practice to ensure that the solution is correct.

Step 6: Verification (Optional but Recommended)

To ensure our solution is correct, we can substitute x = 2 back into the original equation:

(6 - 8x) / 10 = (8 - 6x) / 8 - (10 - 4x) / 4

Substitute x = 2:

(6 - 8(2)) / 10 = (8 - 6(2)) / 8 - (10 - 4(2)) / 4

Simplify each term:

(6 - 16) / 10 = (8 - 12) / 8 - (10 - 8) / 4

-10 / 10 = -4 / 8 - 2 / 4

-1 = -0.5 - 0.5

-1 = -1

Since both sides of the equation are equal, our solution x = 2 is correct. Verification is an important step in problem-solving, as it helps to catch any errors made during the solution process. By substituting the solution back into the original equation, we can confirm that it satisfies the equation and that our answer is accurate.

Conclusion

We have successfully solved the equation (6-8x)/10 = (8-6x)/8 - (10-4x)/4 and found the solution to be x = 2. This process involved several key steps, including finding the LCM, clearing fractions, expanding brackets, combining like terms, isolating the variable, and solving for x. Each step is crucial in simplifying the equation and arriving at the correct solution. Mastering these steps is essential for solving linear equations and other algebraic problems.

By following these steps carefully, you can solve similar equations with confidence. Remember to always verify your solution to ensure accuracy. Algebra is a fundamental branch of mathematics, and solving equations is a core skill that is widely used in various fields. Practice and understanding of these concepts will greatly enhance your mathematical abilities.

The correct answer from the options provided is:

2 (Correct)