Solve: ${ \frac{3}{t-2} = \frac{4}{3t+4} }$

$$

Introduction

Solving equations involving fractions can be a challenging task, especially when the fractions have different denominators. In this article, we will focus on solving the equation $\frac{3}{t-2} = \frac{4}{3t+4}$, which involves fractions with different denominators. We will use algebraic techniques to simplify the equation and solve for the variable $t$.

Understanding the Equation

The given equation is $\frac{3}{t-2} = \frac{4}{3t+4}$. To solve this equation, we need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. The LCM of $t-2$ and $3t+4$ is $(t-2)(3t+4)$.

Step 1: Multiply Both Sides by the LCM

To eliminate the fractions, we multiply both sides of the equation by the LCM, which is $(t-2)(3t+4)$. This gives us:

$$3(3t+4) = 4(t-2)$$

Step 2: Expand and Simplify

We expand and simplify both sides of the equation:

$$9t + 12 = 4t - 8$$

Step 3: Isolate the Variable

To isolate the variable $t$, we need to get all the terms involving $t$ on one side of the equation. We can do this by subtracting $4t$ from both sides of the equation:

$$5t + 12 = -8$$

Step 4: Solve for the Variable

Now, we can solve for the variable $t$ by subtracting $12$ from both sides of the equation:

$$5t = -20$$

Step 5: Final Step

Finally, we can solve for the variable $t$ by dividing both sides of the equation by $5$:

$$t = -4$$

Conclusion

In this article, we solved the equation $\frac{3}{t-2} = \frac{4}{3t+4}$ using algebraic techniques. We eliminated the fractions by multiplying both sides of the equation by the LCM of the denominators and then isolated the variable $t$ by subtracting and adding terms. The final solution is $t = -4$.

Discussion

The equation $\frac{3}{t-2} = \frac{4}{3t+4}$ is a classic example of an equation involving fractions with different denominators. Solving such equations requires careful manipulation of the fractions and algebraic techniques to isolate the variable. In this article, we demonstrated how to solve such equations using the least common multiple (LCM) and algebraic techniques.

Applications

The equation $\frac{3}{t-2} = \frac{4}{3t+4}$ has many applications in mathematics and science. For example, it can be used to model real-world problems involving rates and ratios. In physics, it can be used to describe the motion of objects with varying velocities.

Tips and Tricks

When solving equations involving fractions, it's essential to remember the following tips and tricks:

• Multiply both sides of the equation by the LCM of the denominators to eliminate the fractions.
• Use algebraic techniques to isolate the variable.
• Be careful when subtracting and adding terms to avoid errors.

Final Thoughts

Solving equations involving fractions can be a challenging task, but with practice and patience, it can become second nature. In this article, we demonstrated how to solve the equation $\frac{3}{t-2} = \frac{4}{3t+4}$ using algebraic techniques. We hope that this article has provided valuable insights and tips for solving such equations.

References

• [1] Algebra: A Comprehensive Introduction, Michael Artin
• [2] Calculus: Early Transcendentals, James Stewart
• [3] Mathematics for the Nonmathematician, Morris Kline

Additional Resources

For more information on solving equations involving fractions, we recommend the following resources:

• Khan Academy: Solving Equations with Fractions
• Mathway: Solving Equations with Fractions
• Wolfram Alpha: Solving Equations with Fractions
  

Introduction

Solving equations involving fractions can be a challenging task, but with the right techniques and strategies, it can become a breeze. In this article, we will answer some of the most frequently asked questions about solving equations involving fractions.

Q1: What is the first step in solving an equation involving fractions?

A1: The first step in solving an equation involving fractions is to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q2: How do I find the LCM of the denominators?

A2: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that is common to both lists. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q3: What if the denominators are not factorable?

A3: If the denominators are not factorable, you can use the following trick: multiply both sides of the equation by the product of the denominators. This will eliminate the fractions and allow you to solve for the variable.

Q4: How do I isolate the variable in an equation involving fractions?

A4: To isolate the variable in an equation involving fractions, you need to get all the terms involving the variable on one side of the equation. You can do this by adding or subtracting terms from both sides of the equation.

Q5: What if I have a fraction with a variable in the denominator?

A5: If you have a fraction with a variable in the denominator, you can use the following trick: multiply both sides of the equation by the conjugate of the denominator. The conjugate of a denominator is obtained by changing the sign of the variable in the denominator.

Q6: How do I check my solution to an equation involving fractions?

A6: To check your solution to an equation involving fractions, you need to plug the solution back into the original equation and verify that it is true. If the solution satisfies the original equation, then it is a valid solution.

Q7: What are some common mistakes to avoid when solving equations involving fractions?

A7: Some common mistakes to avoid when solving equations involving fractions include:

• Not eliminating the fractions before solving for the variable
• Not checking the solution to the equation
• Not using the correct technique to isolate the variable

Q8: How can I practice solving equations involving fractions?

A8: 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, using a calculator or computer program to check your solutions.

Q9: What are some real-world applications of solving equations involving fractions?

A9: Solving equations involving fractions has many real-world applications, including:

• Modeling real-world problems involving rates and ratios
• Describing the motion of objects with varying velocities
• Solving problems in finance, economics, and engineering

Q10: How can I improve my skills in solving equations involving fractions?

A10: To improve your skills in solving equations involving fractions, you need to practice regularly and consistently. You can also try working through challenging problems and exercises, and seeking help from a teacher or tutor if you need it.

Conclusion

Solving equations involving fractions can be a challenging task, but with the right techniques and strategies, it can become a breeze. By following the tips and tricks outlined in this article, you can improve your skills in solving equations involving fractions and become a master of algebra.

Additional Resources

For more information on solving equations involving fractions, we recommend the following resources:

• Khan Academy: Solving Equations with Fractions
• Mathway: Solving Equations with Fractions
• Wolfram Alpha: Solving Equations with Fractions

References

• [1] Algebra: A Comprehensive Introduction, Michael Artin
• [2] Calculus: Early Transcendentals, James Stewart
• [3] Mathematics for the Nonmathematician, Morris Kline