Enter The Values For $m, N$, And $p$ That Complete The Difference:$\[ \frac{7}{x} - \frac{3}{2} = \frac{n - M X}{p X} \\]$\[ \begin{array}{l} m = \square \\ n = \square \\ p = \square \\ \end{array} \\]

Introduction

In this article, we will delve into the world of algebra and focus on solving a specific equation involving fractions. The given equation is $\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$. Our goal is to find the values of $m$, $n$, and $p$ that complete the difference. This involves manipulating the equation to isolate the variables and solve for their respective values.

Understanding the Equation

The given equation is a rational equation, which means it involves fractions with variables in the numerator and denominator. To solve this equation, we need to first simplify it by finding a common denominator for the fractions. Once we have a common denominator, we can combine the fractions and then proceed to solve for the variables.

Simplifying the Equation

To simplify the equation, we need to find a common denominator for the fractions. The common denominator for the fractions $\frac{7}{x}$ and $\frac{3}{2}$ is $2x$. We can rewrite the equation as follows:

$\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$

$\frac{14}{2x} - \frac{3x}{2x} = \frac{n - m x}{p x}$

$\frac{14 - 3x^2}{2x} = \frac{n - m x}{p x}$

Combining the Fractions

Now that we have a common denominator, we can combine the fractions on the left-hand side of the equation:

$\frac{14 - 3x^2}{2x} = \frac{n - m x}{p x}$

$\frac{14 - 3x^2}{2x} = \frac{n - m x}{p x}$

Solving for the Variables

To solve for the variables, we need to isolate them on one side of the equation. We can start by multiplying both sides of the equation by $2xp$ to eliminate the fractions:

$2xp \cdot \frac{14 - 3x^2}{2x} = 2xp \cdot \frac{n - m x}{p x}$

$2p(14 - 3x^2) = 2n - 2mx$

Expanding and Simplifying

Now that we have eliminated the fractions, we can expand and simplify the equation:

$28p - 6px^2 = 2n - 2mx$

Rearranging the Terms

To isolate the variables, we need to rearrange the terms in the equation. We can start by moving all the terms involving $x$ to one side of the equation:

$28p - 2n = 2mx - 6px^2$

Factoring Out the Common Term

Now that we have isolated the variables, we can factor out the common term $x$ from the right-hand side of the equation:

$28p - 2n = x(2m - 6px)$

Solving for the Variables

To solve for the variables, we need to set the equation equal to zero. We can do this by setting the left-hand side of the equation equal to zero:

$28p - 2n = 0$

Solving for $m$

Now that we have isolated the variables, we can solve for $m$. We can start by dividing both sides of the equation by $2x$:

$m = \frac{28p - 2n}{2x}$

Solving for $n$

To solve for $n$, we can start by multiplying both sides of the equation by $2x$:

$2n = 28p - 2mx$

Solving for $p$

To solve for $p$, we can start by multiplying both sides of the equation by $2x$:

$2p = \frac{2n - 2mx}{28}$

Conclusion

In this article, we have solved the equation $\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$ and found the values of $m$, $n$, and $p$ that complete the difference. We have used algebraic manipulation to simplify the equation and isolate the variables. The final values of $m$, $n$, and $p$ are:

$m = \frac{28p - 2n}{2x}$

$n = \frac{28p - 2mx}{2}$

$p = \frac{2n - 2mx}{56}$

These values can be used to complete the difference in the given equation.

Discussion

The equation $\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$ is a rational equation that involves fractions with variables in the numerator and denominator. To solve this equation, we need to simplify it by finding a common denominator for the fractions and then proceed to solve for the variables. The final values of $m$, $n$, and $p$ are:

$m = \frac{28p - 2n}{2x}$

$n = \frac{28p - 2mx}{2}$

$p = \frac{2n - 2mx}{56}$

These values can be used to complete the difference in the given equation.

References

• [1] Algebra, 2nd ed. by Michael Artin
• [2] Calculus, 3rd ed. by Michael Spivak
• [3] Linear Algebra and Its Applications, 4th ed. by Gilbert Strang

Glossary

• Rational Equation: An equation that involves fractions with variables in the numerator and denominator.
• Common Denominator: A common multiple of the denominators of two or more fractions.
• Algebraic Manipulation: The process of simplifying and rearranging an equation to isolate the variables.
• Variables: The unknown values in an equation that need to be solved for.
  

Introduction

In our previous article, we solved the equation $\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$ and found the values of $m$, $n$, and $p$ that complete the difference. In this article, we will answer some frequently asked questions related to solving the equation.

Q: What is the common denominator for the fractions in the equation?

A: The common denominator for the fractions in the equation is $2x$.

Q: How do I simplify the equation to isolate the variables?

A: To simplify the equation, you need to find a common denominator for the fractions and then combine them. Once you have a common denominator, you can multiply both sides of the equation by the common denominator to eliminate the fractions.

Q: How do I solve for the variables $m$, $n$, and $p$?

A: To solve for the variables $m$, $n$, and $p$, you need to isolate them on one side of the equation. You can do this by multiplying both sides of the equation by $2xp$ to eliminate the fractions, and then rearranging the terms to isolate the variables.

Q: What are the final values of $m$, $n$, and $p$?

A: The final values of $m$, $n$, and $p$ are:

$m = \frac{28p - 2n}{2x}$

$n = \frac{28p - 2mx}{2}$

$p = \frac{2n - 2mx}{56}$

Q: How do I use the values of $m$, $n$, and $p$ to complete the difference in the equation?

A: To use the values of $m$, $n$, and $p$ to complete the difference in the equation, you can substitute the values into the equation and simplify. This will give you the final solution to the equation.

Q: What are some common mistakes to avoid when solving the equation?

A: Some common mistakes to avoid when solving the equation include:

• Not finding a common denominator for the fractions
• Not multiplying both sides of the equation by the common denominator to eliminate the fractions
• Not isolating the variables on one side of the equation
• Not simplifying the equation to its simplest form

Q: How do I check my solution to the equation?

A: To check your solution to the equation, you can substitute the values of $m$, $n$, and $p$ back into the original equation and simplify. If the equation is true, then your solution is correct.

Conclusion

In this article, we have answered some frequently asked questions related to solving the equation $\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$. We have provided step-by-step instructions on how to simplify the equation, isolate the variables, and solve for the values of $m$, $n$, and $p$. We have also provided some common mistakes to avoid and tips on how to check your solution to the equation.

Discussion

Solving the equation $\frac{7}{x} - \frac{3}{2} = \frac{n - m x}{p x}$ requires careful attention to detail and a thorough understanding of algebraic manipulation. By following the steps outlined in this article, you can solve the equation and find the values of $m$, $n$, and $p$ that complete the difference.

References

• [1] Algebra, 2nd ed. by Michael Artin
• [2] Calculus, 3rd ed. by Michael Spivak
• [3] Linear Algebra and Its Applications, 4th ed. by Gilbert Strang

Glossary

• Rational Equation: An equation that involves fractions with variables in the numerator and denominator.
• Common Denominator: A common multiple of the denominators of two or more fractions.
• Algebraic Manipulation: The process of simplifying and rearranging an equation to isolate the variables.
• Variables: The unknown values in an equation that need to be solved for.