Solving For K In The Equation 2(kx - N) = -28/15x - 36/19 With No Solution

In this article, we will delve into the process of finding the value of k in the equation 2(kx - n) = -28/15 x - 36/19, given that k and n are constants, n > 1, and the equation has no solution. This problem involves algebraic manipulation and an understanding of the conditions under which a linear equation has no solution. We will break down the steps involved in solving this problem, providing a clear and concise explanation for each step. Understanding the nuances of linear equations and their solutions is crucial in mathematics, and this article aims to equip you with the necessary tools and knowledge to tackle similar problems.

The given equation is 2(kx - n) = -28/15 x - 36/19. We are told that this equation has no solution, which means that there is no value of x that can satisfy the equation. This condition occurs when the coefficients of x on both sides of the equation are equal, but the constant terms are different. In simpler terms, the lines represented by the two sides of the equation are parallel but not overlapping. To solve for k, we need to first expand the equation and then compare the coefficients of x and the constant terms. By setting the coefficients of x equal to each other, we can find the value of k that makes the equation have no solution. This involves careful algebraic manipulation and a clear understanding of the conditions for a linear equation to have no solution.

Expanding the Equation

Our first step is to expand the left side of the equation: 2(kx - n) = -28/15 x - 36/19. Distributing the 2 across the terms inside the parenthesis gives us 2_kx_ - 2_n_ = -28/15 x - 36/19. This expansion is crucial as it separates the x terms and the constant terms, making it easier to compare coefficients later on. By expanding the equation, we are essentially rewriting it in a form that allows us to clearly see the relationship between the variables and constants. This step is fundamental in solving algebraic equations and is a common technique used to simplify and rearrange equations.

Rearranging the Equation

Next, we want to group the x terms on one side and the constant terms on the other. To do this, we can add 28/15 x to both sides and add 2_n_ to both sides. This gives us 2_kx_ + 28/15 x = 2_n_ - 36/19. This rearrangement is a key step in solving the equation as it isolates the x terms and the constant terms, making it easier to compare them. By rearranging the equation, we are essentially setting up the equation in a form that allows us to identify the conditions for no solution. This step is a common technique used in algebra to simplify and solve equations.

Combining x Terms

Now, we can factor out x from the left side of the equation: x(2_k_ + 28/15) = 2_n_ - 36/19. This step is important because it allows us to clearly see the coefficient of x, which is (2_k_ + 28/15). By factoring out x, we are essentially isolating the term that determines the slope of the line represented by the equation. This is crucial in determining the conditions for no solution, as the coefficients of x on both sides of the equation must be equal for the equation to have no solution.

Condition for No Solution

For the equation to have no solution, the coefficients of x on both sides must be equal, and the constant terms must be different. This means that 2_k_ + 28/15 must be equal to 0, and 2_n_ - 36/19 must not be equal to 0. The condition for no solution arises from the fact that parallel lines have the same slope but different y-intercepts. In the context of linear equations, this translates to the coefficients of x being equal, but the constant terms being different. Understanding this condition is crucial in solving problems where we need to find the values of parameters that make an equation have no solution.

Solving for k

Setting the coefficient of x to zero, we have 2_k_ + 28/15 = 0. To solve for k, we can subtract 28/15 from both sides, which gives us 2_k_ = -28/15. Then, we divide both sides by 2 to get k = -28/15 ÷ 2 = -28/15 × 1/2 = -14/15. This is the value of k that makes the coefficients of x equal on both sides of the equation. By solving for k, we are essentially finding the value that makes the lines represented by the two sides of the equation parallel. This step is a crucial part of the solution process and requires careful algebraic manipulation.

Verifying the Constant Terms

We also need to ensure that the constant terms are different. Since n > 1, we have 2_n_ > 2. Therefore, 2_n_ - 36/19 > 2 - 36/19 = (38 - 36)/19 = 2/19, which is not equal to zero. This confirms that the constant terms are indeed different, which is a necessary condition for the equation to have no solution. Verifying the constant terms is an important step in the solution process as it ensures that the lines represented by the two sides of the equation are not only parallel but also not overlapping. This step completes the verification process and confirms that the value of k we found indeed makes the equation have no solution.

Therefore, the value of k that makes the equation 2(kx - n) = -28/15 x - 36/19 have no solution is -14/15. This solution is obtained by equating the coefficients of x on both sides of the equation and solving for k. The condition n > 1 ensures that the constant terms are different, which is a necessary condition for the equation to have no solution. By following the step-by-step solution process, we have clearly demonstrated how to solve this problem and arrive at the correct answer. This problem highlights the importance of understanding the conditions for a linear equation to have no solution and the algebraic techniques used to solve such problems.

In this article, we have walked through the process of solving for k in the equation 2(kx - n) = -28/15 x - 36/19, given that the equation has no solution and n > 1. We expanded the equation, rearranged the terms, and identified the condition for no solution, which is that the coefficients of x must be equal while the constant terms are different. By setting the coefficients of x equal to each other and solving for k, we found that k = -14/15. We also verified that the constant terms are different, ensuring that the equation indeed has no solution. This problem provides valuable insights into the properties of linear equations and the conditions under which they have no solution. The techniques used in this solution, such as expanding, rearranging, and comparing coefficients, are fundamental in algebra and are applicable to a wide range of mathematical problems. Understanding these concepts and techniques is crucial for success in mathematics and related fields.