Solve For $k$.$\frac{1}{k+1} = \frac{2k}{4}$There May Be 1 Or 2 Solutions.$k =$ $\square$ Or $k =$ $\square$

Introduction

In this article, we will delve into solving a linear equation involving fractions. The equation given is $\frac{1}{k+1} = \frac{2k}{4}$, and we are tasked with finding the value of $k$. This equation may have one or two solutions, and we will explore both possibilities.

Step 1: Simplify the Equation

To begin solving the equation, we can simplify the right-hand side by dividing both the numerator and denominator by their greatest common divisor, which is 2. This gives us $\frac{1}{k+1} = \frac{k}{2}$.

Step 2: Cross-Multiply

Next, we can cross-multiply to eliminate the fractions. This involves multiplying both sides of the equation by the denominators, which are $k+1$ and 2. This gives us $2 = k(k+1)$.

Step 3: Expand and Simplify

We can now expand the right-hand side of the equation by multiplying $k$ and $k+1$. This gives us $2 = k^2 + k$.

Step 4: Rearrange the Equation

To make it easier to solve, we can rearrange the equation by subtracting 2 from both sides. This gives us $0 = k^2 + k - 2$.

Step 5: Factor the Quadratic Equation

The equation $0 = k^2 + k - 2$ is a quadratic equation, and we can factor it by finding two numbers whose product is -2 and whose sum is 1. These numbers are 2 and -1, so we can factor the equation as $0 = (k+2)(k-1)$.

Step 6: Solve for $k$

Now that we have factored the equation, we can set each factor equal to zero and solve for $k$. This gives us two possible solutions: $k+2=0$ and $k-1=0$. Solving for $k$ in each case, we get $k=-2$ and $k=1$.

Conclusion

In this article, we have solved the equation $\frac{1}{k+1} = \frac{2k}{4}$ and found two possible solutions: $k=-2$ and $k=1$. These solutions satisfy the original equation, and we can verify this by plugging them back into the equation.

Final Answer

The final answer is: $\boxed{-2}$ or $\boxed{1}$

Introduction

In our previous article, we solved the equation $\frac{1}{k+1} = \frac{2k}{4}$ and found two possible solutions: $k=-2$ and $k=1$. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is the first step in solving the equation $\frac{1}{k+1} = \frac{2k}{4}$?

A: The first step in solving the equation is to simplify the right-hand side by dividing both the numerator and denominator by their greatest common divisor, which is 2. This gives us $\frac{1}{k+1} = \frac{k}{2}$.

Q: Why do we need to cross-multiply in this equation?

A: We need to cross-multiply to eliminate the fractions. This involves multiplying both sides of the equation by the denominators, which are $k+1$ and 2. This gives us $2 = k(k+1)$.

Q: How do we expand and simplify the equation $2 = k(k+1)$?

A: We can expand the right-hand side of the equation by multiplying $k$ and $k+1$. This gives us $2 = k^2 + k$.

Q: Why do we need to rearrange the equation $2 = k^2 + k$?

A: We need to rearrange the equation by subtracting 2 from both sides. This gives us $0 = k^2 + k - 2$.

Q: How do we factor the quadratic equation $0 = k^2 + k - 2$?

A: We can factor the quadratic equation by finding two numbers whose product is -2 and whose sum is 1. These numbers are 2 and -1, so we can factor the equation as $0 = (k+2)(k-1)$.

Q: What are the two possible solutions to the equation $\frac{1}{k+1} = \frac{2k}{4}$?

A: The two possible solutions to the equation are $k=-2$ and $k=1$.

Q: How do we verify that these solutions satisfy the original equation?

A: We can verify that these solutions satisfy the original equation by plugging them back into the equation. For example, we can plug in $k=-2$ and $k=1$ into the equation $\frac{1}{k+1} = \frac{2k}{4}$ and check if the equation holds true.

Q: What is the final answer to the equation $\frac{1}{k+1} = \frac{2k}{4}$?

A: The final answer to the equation is $\boxed{-2}$ or $\boxed{1}$.

Conclusion

In this article, we have answered some frequently asked questions about solving the equation $\frac{1}{k+1} = \frac{2k}{4}$. We have covered the steps involved in solving the equation, including simplifying, cross-multiplying, expanding, and factoring. We have also verified that the two possible solutions satisfy the original equation.

Final Answer

The final answer is: $\boxed{-2}$ or $\boxed{1}$