What Are The Solutions To The Equation $\frac{w}{2w-3}=\frac{4}{w}$?A. $w=-6$ And $w=-2$ B. $w=0$, $w=2$, And $w=6$ C. $w=0$ And $w=\frac{3}{2}$ D. $w=2$ And

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Introduction

Solving equations involving fractions can be a challenging task, especially when the fractions are not simplified. In this article, we will explore the solutions to the equation $\frac{w}{2w-3}=\frac{4}{w}$, which involves a rational expression. We will use algebraic techniques to simplify the equation and find the values of $w$ that satisfy the equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by cross-multiplying both sides of the equation. This will eliminate the fractions and make it easier to solve for $w$. Cross-multiplying is a technique used to eliminate fractions by multiplying both sides of the equation by the denominators.

\frac{w}{2w-3}=\frac{4}{w}

Cross-multiplying both sides of the equation gives us:

w^2 = 4(2w-3)

Step 2: Expand and Simplify

Next, we can expand and simplify the equation by distributing the $4$ on the right-hand side of the equation.

w^2 = 8w - 12

Step 3: Rearrange the Equation

To make it easier to solve for $w$, we can rearrange the equation by moving all the terms to one side of the equation.

w^2 - 8w + 12 = 0

Step 4: Factor the Quadratic Equation

The equation is a quadratic equation, and we can try to factor it to find the values of $w$ that satisfy the equation. Factoring a quadratic equation involves finding two binomials whose product is equal to the quadratic expression.

(w - 2)(w - 6) = 0

Step 5: Solve for $w$

To find the values of $w$ that satisfy the equation, we can set each factor equal to zero and solve for $w$.

w - 2 = 0 \quad \text{or} \quad w - 6 = 0

Solving for $w$ gives us:

w = 2 \quad \text{or} \quad w = 6

Conclusion

In this article, we have explored the solutions to the equation $\frac{w}{2w-3}=\frac{4}{w}$. We used algebraic techniques to simplify the equation and find the values of $w$ that satisfy the equation. The solutions to the equation are $w = 2$ and $w = 6$.

Discussion

The equation $\frac{w}{2w-3}=\frac{4}{w}$ is a rational equation, and solving it involves simplifying the equation and finding the values of $w$ that satisfy the equation. The solutions to the equation are $w = 2$ and $w = 6$. These values of $w$ make the original equation true, and they satisfy the equation.

Final Answer

The final answer is $\boxed{w = 2 \text{ and } w = 6}$.

Comparison of Solutions

Let's compare the solutions to the equation with the options provided.

• Option A: $w = -6$ and $w = -2$
• Option B: $w = 0$, $w = 2$, and $w = 6$
• Option C: $w = 0$ and $w = \frac{3}{2}$
• Option D: $w = 2$ and $w = 6$

The correct solution is Option B: $w = 0$, $w = 2$, and $w = 6$.

Conclusion

In this article, we have explored the solutions to the equation $\frac{w}{2w-3}=\frac{4}{w}$. We used algebraic techniques to simplify the equation and find the values of $w$ that satisfy the equation. The solutions to the equation are $w = 0$, $w = 2$, and $w = 6$.

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Introduction

In our previous article, we explored the solutions to the equation $\frac{w}{2w-3}=\frac{4}{w}$. We used algebraic techniques to simplify the equation and find the values of $w$ that satisfy the equation. In this article, we will answer some frequently asked questions (FAQs) about solving the equation.

Q: What is the first step in solving the equation $\frac{w}{2w-3}=\frac{4}{w}$?

A: The first step in solving the equation is to cross-multiply both sides of the equation. This will eliminate the fractions and make it easier to solve for $w$.

Q: Why do we need to cross-multiply both sides of the equation?

A: We need to cross-multiply both sides of the equation to eliminate the fractions. This will make it easier to solve for $w$ and find the values that satisfy the equation.

Q: What is the next step after cross-multiplying both sides of the equation?

A: After cross-multiplying both sides of the equation, we need to expand and simplify the equation. This will involve distributing the $4$ on the right-hand side of the equation.

Q: How do we expand and simplify the equation?

A: To expand and simplify the equation, we need to distribute the $4$ on the right-hand side of the equation. This will give us a quadratic equation that we can solve for $w$.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to factor the quadratic equation and solve for $w$. This will give us the values of $w$ that satisfy the equation.

Q: What are the solutions to the equation $\frac{w}{2w-3}=\frac{4}{w}$?

A: The solutions to the equation are $w = 0$, $w = 2$, and $w = 6$. These values of $w$ make the original equation true and satisfy the equation.

Q: Why are the solutions $w = 0$, $w = 2$, and $w = 6$?

A: The solutions $w = 0$, $w = 2$, and $w = 6$ are the values of $w$ that make the original equation true. These values satisfy the equation and make the equation true.

Q: Can we use other methods to solve the equation?

A: Yes, we can use other methods to solve the equation. However, the method we used in this article is a common and effective method for solving rational equations.

Q: What are some common mistakes to avoid when solving rational equations?

A: Some common mistakes to avoid when solving rational equations include:

• Not cross-multiplying both sides of the equation
• Not expanding and simplifying the equation
• Not factoring the quadratic equation
• Not checking the solutions to make sure they satisfy the equation

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about solving the equation $\frac{w}{2w-3}=\frac{4}{w}$. We have covered the steps involved in solving the equation, the solutions to the equation, and some common mistakes to avoid when solving rational equations.

Final Answer

The final answer is $\boxed{w = 0, w = 2, \text{ and } w = 6}$.

Comparison of Solutions

Let's compare the solutions to the equation with the options provided.

• Option A: $w = -6$ and $w = -2$
• Option B: $w = 0$, $w = 2$, and $w = 6$
• Option C: $w = 0$ and $w = \frac{3}{2}$
• Option D: $w = 2$ and $w = 6$

The correct solution is Option B: $w = 0$, $w = 2$, and $w = 6$.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about solving the equation $\frac{w}{2w-3}=\frac{4}{w}$. We have covered the steps involved in solving the equation, the solutions to the equation, and some common mistakes to avoid when solving rational equations.