Solve The Equation:${ \frac{5}{3v - 2} = \frac{7}{4v} }$

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Introduction

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In this article, we will delve into the world of algebra and solve a complex equation involving fractions. The equation we will be working with is $\frac{5}{3v - 2} = \frac{7}{4v}$. Our goal is to isolate the variable $v$ and find its value. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Cross-Multiply

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The first step in solving this equation is to cross-multiply. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This will eliminate the fractions and make it easier to work with the equation.

$\frac{5}{3v - 2} = \frac{7}{4v}$

Cross-multiplying gives us:

$5 \cdot 4v = 7 \cdot (3v - 2)$

Step 2: Expand and Simplify

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Next, we need to expand and simplify the equation. This involves multiplying out the terms and combining like terms.

$20v = 21v - 14$

Step 3: Isolate the Variable

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Now, we need to isolate the variable $v$. To do this, we will move all the terms involving $v$ to one side of the equation and the constant terms to the other side.

$20v - 21v = -14$

Step 4: Solve for $v$

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Finally, we can solve for $v$ by dividing both sides of the equation by the coefficient of $v$.

$-v = -14$

$v = 14$

Conclusion

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In this article, we have solved the equation $\frac{5}{3v - 2} = \frac{7}{4v}$ by cross-multiplying, expanding and simplifying, isolating the variable, and solving for $v$. We have shown that the value of $v$ is 14.

Tips and Tricks

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• When solving equations involving fractions, it's often helpful to cross-multiply to eliminate the fractions.
• Make sure to expand and simplify the equation after cross-multiplying.
• Isolate the variable by moving all the terms involving the variable to one side of the equation and the constant terms to the other side.
• Finally, solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Real-World Applications

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Solving equations involving fractions has many real-world applications. For example, in physics, equations involving fractions are used to describe the motion of objects. In finance, equations involving fractions are used to calculate interest rates and investment returns. In engineering, equations involving fractions are used to design and optimize systems.

Common Mistakes

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• Failing to cross-multiply when solving equations involving fractions.
• Not expanding and simplifying the equation after cross-multiplying.
• Not isolating the variable by moving all the terms involving the variable to one side of the equation and the constant terms to the other side.
• Not solving for the variable by dividing both sides of the equation by the coefficient of the variable.

Final Thoughts

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Solving equations involving fractions requires patience and attention to detail. By following the steps outlined in this article, you can solve even the most complex equations involving fractions. Remember to cross-multiply, expand and simplify, isolate the variable, and solve for the variable. With practice and experience, you will become proficient in solving equations involving fractions and be able to apply this skill to real-world problems.

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Introduction

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In our previous article, we solved the equation $\frac{5}{3v - 2} = \frac{7}{4v}$ by cross-multiplying, expanding and simplifying, isolating the variable, and solving for $v$. In this article, we will answer some of the most frequently asked questions about solving equations involving fractions.

Q: What is the first step in solving an equation involving fractions?

A: The first step in solving an equation involving fractions is to cross-multiply. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa.

Q: Why do I need to cross-multiply when solving an equation involving fractions?

A: Cross-multiplying eliminates the fractions and makes it easier to work with the equation. It also allows us to use the properties of equality to solve for the variable.

Q: What is the difference between cross-multiplying and multiplying both sides of the equation by a common denominator?

A: Cross-multiplying involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. Multiplying both sides of the equation by a common denominator involves multiplying both sides by the product of the denominators.

Q: How do I know when to use cross-multiplication and when to use multiplying both sides by a common denominator?

A: You should use cross-multiplication when the equation involves fractions with different denominators. You should use multiplying both sides by a common denominator when the equation involves fractions with the same denominator.

Q: What is the next step after cross-multiplying?

A: After cross-multiplying, you should expand and simplify the equation. This involves multiplying out the terms and combining like terms.

Q: Why do I need to expand and simplify the equation?

A: Expanding and simplifying the equation makes it easier to work with and helps to eliminate any unnecessary terms.

Q: How do I know when I have finished expanding and simplifying the equation?

A: You have finished expanding and simplifying the equation when there are no more terms to combine and the equation is in its simplest form.

Q: What is the next step after expanding and simplifying the equation?

A: After expanding and simplifying the equation, you should isolate the variable. This involves moving all the terms involving the variable to one side of the equation and the constant terms to the other side.

Q: Why do I need to isolate the variable?

A: Isolating the variable makes it easier to solve for the variable and ensures that the equation is in its simplest form.

Q: How do I know when I have finished isolating the variable?

A: You have finished isolating the variable when all the terms involving the variable are on one side of the equation and the constant terms are on the other side.

Q: What is the final step in solving an equation involving fractions?

A: The final step in solving an equation involving fractions is to solve for the variable. This involves dividing both sides of the equation by the coefficient of the variable.

Q: Why do I need to solve for the variable?

A: Solving for the variable gives you the value of the variable and allows you to use the equation to make predictions or solve real-world problems.

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

A: Some common mistakes to avoid when solving equations involving fractions include failing to cross-multiply, not expanding and simplifying the equation, not isolating the variable, and not solving for the variable.

Q: How can I practice solving equations involving fractions?

A: You can practice solving equations involving fractions by working through examples and exercises in a textbook or online resource. You can also try solving equations involving fractions on your own and checking your work with a calculator or online tool.

Conclusion

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Solving equations involving fractions requires patience and attention to detail. By following the steps outlined in this article, you can solve even the most complex equations involving fractions. Remember to cross-multiply, expand and simplify, isolate the variable, and solve for the variable. With practice and experience, you will become proficient in solving equations involving fractions and be able to apply this skill to real-world problems.