Solve For { M $} . . . { \frac{m}{5} = \frac{m-6}{4} \}

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Introduction

Solving for $m$ in the given equation $\frac{m}{5} = \frac{m-6}{4}$ is a fundamental problem in algebra that requires careful manipulation of the equation to isolate the variable $m$. In this article, we will walk through the step-by-step process of solving for $m$ and provide a clear explanation of the mathematical concepts involved.

Understanding the Equation

The given equation is a rational equation, which means it contains fractions with variables in the numerator and denominator. The equation is $\frac{m}{5} = \frac{m-6}{4}$. Our goal is to solve for $m$, which means we need to isolate $m$ on one side of the equation.

Step 1: Cross-Multiply

To solve for $m$, we can start by cross-multiplying the two fractions. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This gives us:

$$m \cdot 4 = (m-6) \cdot 5$$

Step 2: Expand and Simplify

Next, we can expand and simplify the equation by multiplying the terms on the right-hand side:

$$4m = 5m - 30$$

Step 3: Isolate $m$

Now, we can isolate $m$ by subtracting $5m$ from both sides of the equation:

$$-m = -30$$

Step 4: Solve for $m$

Finally, we can solve for $m$ by dividing both sides of the equation by $-1$:

$$m = 30$$

Conclusion

In this article, we have walked through the step-by-step process of solving for $m$ in the given equation $\frac{m}{5} = \frac{m-6}{4}$. We have used the techniques of cross-multiplication, expansion, and simplification to isolate $m$ on one side of the equation. The final solution is $m = 30$.

Tips and Tricks

• When solving rational equations, it's essential to cross-multiply to eliminate the fractions.
• Be careful when expanding and simplifying the equation to avoid making mistakes.
• Isolate the variable $m$ on one side of the equation to ensure a correct solution.

Real-World Applications

Solving for $m$ in rational equations has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the equation $\frac{m}{5} = \frac{m-6}{4}$ can be used to model the motion of an object with a variable mass. In engineering, the equation can be used to design systems with variable parameters. In economics, the equation can be used to model the behavior of economic systems with variable variables.

Common Mistakes

• Failing to cross-multiply when solving rational equations.
• Making mistakes when expanding and simplifying the equation.
• Failing to isolate the variable $m$ on one side of the equation.

Final Thoughts

Solving for $m$ in rational equations requires careful attention to detail and a thorough understanding of the mathematical concepts involved. By following the step-by-step process outlined in this article, you can confidently solve for $m$ in a variety of rational equations. Remember to always cross-multiply, expand and simplify, and isolate the variable $m$ on one side of the equation to ensure a correct solution.

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Introduction

In our previous article, we walked through the step-by-step process of solving for $m$ in the equation $\frac{m}{5} = \frac{m-6}{4}$. In this article, we will answer some of the most frequently asked questions about solving for $m$ in rational equations.

Q: What is the first step in solving for $m$ in a rational equation?

A: The first step in solving for $m$ in a rational equation 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 is cross-multiplication necessary when solving for $m$ in a rational equation?

A: Cross-multiplication is necessary when solving for $m$ in a rational equation because it allows us to eliminate the fractions and work with a simpler equation.

Q: What is the difference between cross-multiplication and multiplication?

A: Cross-multiplication is a specific technique used to eliminate fractions in rational equations. Multiplication, on the other hand, is a more general operation that can be used to simplify equations.

Q: How do I know when to cross-multiply in a rational equation?

A: You should cross-multiply in a rational equation when you see two fractions with variables in the numerator and denominator. This is a clear indication that you need to eliminate the fractions to solve for the variable.

Q: What if I have a rational equation with more than two fractions?

A: If you have a rational equation with more than two fractions, you can still use cross-multiplication to eliminate the fractions. However, you may need to use additional techniques, such as factoring or simplifying, to solve for the variable.

Q: Can I use cross-multiplication to solve for $m$ in an equation with variables on both sides?

A: Yes, you can use cross-multiplication to solve for $m$ in an equation with variables on both sides. However, you will need to be careful to isolate the variable $m$ on one side of the equation.

Q: What if I make a mistake when cross-multiplying?

A: If you make a mistake when cross-multiplying, you may end up with an incorrect solution. To avoid this, make sure to double-check your work and use a calculator or other tool to verify your answer.

Q: Can I use a calculator to solve for $m$ in a rational equation?

A: Yes, you can use a calculator to solve for $m$ in a rational equation. However, keep in mind that a calculator may not always provide the most accurate or efficient solution.

Q: How do I know if my solution is correct?

A: To determine if your solution is correct, make sure to check your work and verify that the solution satisfies the original equation.

Q: What if I get stuck when solving for $m$ in a rational equation?

A: If you get stuck when solving for $m$ in a rational equation, try breaking down the problem into smaller steps or seeking help from a teacher or tutor.

Conclusion

Solving for $m$ in rational equations can be a challenging task, but with practice and patience, you can become proficient in using cross-multiplication and other techniques to solve for the variable. Remember to always double-check your work and verify your answer to ensure accuracy.

Tips and Tricks

• Make sure to cross-multiply carefully to avoid mistakes.
• Use a calculator or other tool to verify your answer.
• Break down complex problems into smaller steps.
• Seek help from a teacher or tutor if you get stuck.

Real-World Applications

Solving for $m$ in rational equations has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the equation $\frac{m}{5} = \frac{m-6}{4}$ can be used to model the motion of an object with a variable mass. In engineering, the equation can be used to design systems with variable parameters. In economics, the equation can be used to model the behavior of economic systems with variable variables.

Common Mistakes

• Failing to cross-multiply when solving rational equations.
• Making mistakes when expanding and simplifying the equation.
• Failing to isolate the variable $m$ on one side of the equation.

Final Thoughts

Solving for $m$ in rational equations requires careful attention to detail and a thorough understanding of the mathematical concepts involved. By following the step-by-step process outlined in this article and practicing regularly, you can become proficient in using cross-multiplication and other techniques to solve for the variable. Remember to always double-check your work and verify your answer to ensure accuracy.