Not multiplying both sides of the equation by the LCM

Oct 13, 2025 by sorusgentrification.com 107 views

Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The equation we will be solving is $\frac{2k+7}{5} - \frac{k+4}{2} = 3\left(\frac{k-6}{4}\right)$ . This equation involves fractions, variables, and constants, making it a challenging problem for even the most experienced mathematicians. However, with a clear understanding of the steps involved, we can break down the solution into manageable parts and arrive at the final answer.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions in the equation, we need to multiply both sides by the least common multiple (LCM) of the denominators. In this case, the LCM of 5, 2, and 4 is 20. Multiplying both sides by 20, we get:

20\left(\frac{2k+7}{5} - \frac{k+4}{2}\right) = 20\left(3\left(\frac{k-6}{4}\right)\right)

Step 2: Distribute the Multiplication

Now that we have multiplied both sides by 20, we can distribute the multiplication to each term inside the parentheses:

20\left(\frac{2k+7}{5}\right) - 20\left(\frac{k+4}{2}\right) = 20\left(3\left(\frac{k-6}{4}\right)\right)

This simplifies to:

4(2k+7) - 10(k+4) = 15(k-6)

Step 3: Simplify the Equation

Now that we have distributed the multiplication, we can simplify the equation by combining like terms:

8k+28-10k-40 = 15k-90

This simplifies to:

-2k-12 = 15k-90

Step 4: Isolate the Variable

To isolate the variable $k$ , we need to get all the terms involving $k$ on one side of the equation. We can do this by adding $2k$ to both sides of the equation:

-12 = 17k-90

Step 5: Add 90 to Both Sides

Next, we need to add 90 to both sides of the equation to get rid of the constant term on the right-hand side:

78 = 17k

Step 6: Divide Both Sides by 17

Finally, we can divide both sides of the equation by 17 to solve for $k$ :

k = \frac{78}{17}

Conclusion

In this article, we have solved a complex equation step by step. We started by multiplying both sides by the least common multiple (LCM) of the denominators, then distributed the multiplication, simplified the equation, isolated the variable, added 90 to both sides, and finally divided both sides by 17 to arrive at the final answer. With practice and patience, anyone can master the art of solving complex equations like this one.

Final Answer

$The final answer is $\boxed{\frac{78}{17}}$.<br/> **Solving the Equation: A Step-by-Step Guide**$

Q&A: Frequently Asked Questions

Q: What is the least common multiple (LCM) of 5, 2, and 4?

A: The least common multiple (LCM) of 5, 2, and 4 is 20.

Q: Why do we need to multiply both sides of the equation by the LCM?

A: We need to multiply both sides of the equation by the LCM to eliminate the fractions in the equation. This makes it easier to solve for the variable.

Q: What is the difference between the LCM and the greatest common divisor (GCD)?

A: The least common multiple (LCM) is the smallest multiple that is a common multiple of two or more numbers, while the greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Q: How do we distribute the multiplication in the equation?

A: We distribute the multiplication by multiplying each term inside the parentheses by the number outside the parentheses.

Q: What is the purpose of simplifying the equation?

A: Simplifying the equation helps to make it easier to solve for the variable by combining like terms.

Q: How do we isolate the variable in the equation?

A: We isolate the variable by getting all the terms involving the variable on one side of the equation.

Q: What is the final answer to the equation?

A: The final answer to the equation is $\boxed{\frac{78}{17}}$ .

Common Mistakes to Avoid

Not multiplying both sides of the equation by the LCM: Failing to eliminate the fractions in the equation can make it difficult to solve for the variable.
Not distributing the multiplication: Failing to distribute the multiplication can lead to incorrect solutions.
Not simplifying the equation: Failing to simplify the equation can make it difficult to solve for the variable.
Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.

Tips and Tricks

Use a calculator to check your answers: Using a calculator can help you verify that your solutions are correct.
Check your work: Double-checking your work can help you catch any mistakes you may have made.
Practice, practice, practice: The more you practice solving equations, the more comfortable you will become with the steps involved.

Conclusion

In this article, we have provided a step-by-step guide to solving a complex equation. We have also answered frequently asked questions and provided tips and tricks to help you master the art of solving equations. With practice and patience, anyone can become proficient in solving equations like this one.