If $6x = 42$ And $xk = 2$, What Is The Value Of $k$?A. 5 B. 7 C. $\frac{1}{6}$ D. $\frac{2}{7}$ E. $\frac{1}{7}$

Solving for the Unknown: A Step-by-Step Guide to Finding the Value of $k$

In this article, we will delve into the world of algebra and explore a simple yet intriguing problem. Given two equations, $6x = 42$ and $xk = 2$, we are tasked with finding the value of $k$. This problem may seem straightforward, but it requires a clear understanding of algebraic manipulation and a step-by-step approach to arrive at the solution.

Before we dive into the solution, let's break down the problem and understand what is being asked. We have two equations:

1. $6x = 42$
2. $xk = 2$

Our goal is to find the value of $k$. To do this, we need to manipulate the equations and isolate the variable $k$.

Let's start by solving the first equation for $x$. We can do this by dividing both sides of the equation by 6:

$$\frac{6x}{6} = \frac{42}{6}$$

This simplifies to:

$$x = 7$$

Now that we have the value of $x$, we can substitute it into the second equation to solve for $k$.

Substituting $x = 7$ into the second equation, we get:

$$7k = 2$$

To solve for $k$, we need to isolate the variable $k$. We can do this by dividing both sides of the equation by 7:

$$\frac{7k}{7} = \frac{2}{7}$$

This simplifies to:

$$k = \frac{2}{7}$$

In conclusion, we have successfully solved for the value of $k$ using the given equations. By following a step-by-step approach and manipulating the equations, we arrived at the solution:

$$k = \frac{2}{7}$$

This problem demonstrates the importance of algebraic manipulation and the need to isolate the variable of interest. By breaking down the problem into smaller steps and using algebraic techniques, we can arrive at the solution and gain a deeper understanding of the underlying mathematics.

The correct answer is:

D. $\frac{2}{7}$

If you are struggling with algebra or need additional practice, here are some tips and resources to help you improve:

• Practice, practice, practice: The more you practice, the more comfortable you will become with algebraic manipulation.
• Use online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer a wealth of resources and practice problems to help you improve your algebra skills.
• Seek help: Don't be afraid to ask for help if you are struggling with a particular concept or problem. Reach out to a teacher, tutor, or classmate for support.

By following these tips and resources, you can improve your algebra skills and become more confident in your ability to solve problems like this one.
Frequently Asked Questions: Solving for the Unknown

A: The main concept behind solving for the unknown in this problem is algebraic manipulation. We use algebraic techniques such as substitution and division to isolate the variable of interest, which in this case is $k$.

A: We need to solve for $x$ before solving for $k$ because the value of $x$ is used in the second equation to solve for $k$. By solving for $x$ first, we can substitute its value into the second equation and then solve for $k$.

A: No, we cannot solve for $k$ without solving for $x$. The value of $x$ is necessary to substitute into the second equation and solve for $k$.

A: Solving for $x$ involves finding the value of $x$ that satisfies the first equation, while solving for $k$ involves finding the value of $k$ that satisfies the second equation. In this problem, we first solve for $x$ and then use its value to solve for $k$.

A: We know which equation to solve first by looking at the problem and identifying the variables involved. In this case, we are given two equations, and we need to solve for $k$. We start by solving for $x$ because its value is used in the second equation to solve for $k$.

A: Yes, we can use other algebraic techniques to solve for $k$. For example, we could use the method of elimination to eliminate the variable $x$ and solve for $k$. However, in this problem, we use substitution and division to solve for $k$.

A: Algebraic manipulation is essential in solving for the unknown because it allows us to isolate the variable of interest and find its value. By using algebraic techniques such as substitution and division, we can solve for the unknown and gain a deeper understanding of the underlying mathematics.

A: The concepts learned in this problem can be applied to real-world situations where we need to solve for unknown variables. For example, in physics, we may need to solve for the velocity of an object given its acceleration and time. In economics, we may need to solve for the demand for a product given its price and income. By applying the concepts learned in this problem, we can solve for unknown variables and make informed decisions in a variety of real-world situations.

A: Some common mistakes to avoid when solving for the unknown include:

• Not following the order of operations
• Not isolating the variable of interest
• Not checking the solution for reasonableness
• Not using algebraic techniques such as substitution and division

By avoiding these common mistakes, we can ensure that our solutions are accurate and reliable.