Solve For Z Z Z . 1 Z − 2 = 5 2 Z − 1 5 \frac{1}{z} - 2 = \frac{5}{2z} - \frac{1}{5} Z 1 ​ − 2 = 2 Z 5 ​ − 5 1 ​

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics, and it is essential to understand how to manipulate and solve equations to find the value of unknown variables. In this article, we will focus on solving a specific equation involving fractions and variables. The given equation is $\frac{1}{z} - 2 = \frac{5}{2z} - \frac{1}{5}$, and our goal is to solve for the variable $z$.

Understanding the Equation

Before we start solving the equation, let's take a closer look at it. The equation involves fractions and variables, and it may seem intimidating at first. However, with a step-by-step approach, we can break down the equation and solve for $z$. The equation can be rewritten as:

$\frac{1}{z} - 2 = \frac{5}{2z} - \frac{1}{5}$

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM is $10z$. Multiplying both sides by $10z$ gives us:

$10z \left( \frac{1}{z} - 2 \right) = 10z \left( \frac{5}{2z} - \frac{1}{5} \right)$

Step 3: Simplify the Equation

Now that we have eliminated the fractions, we can simplify the equation. Expanding the left-hand side of the equation gives us:

$10z \left( \frac{1}{z} \right) - 20z = 10z \left( \frac{5}{2z} \right) - 2z$

Simplifying further, we get:

$10 - 20z = 25 - 2z$

Step 4: Isolate the Variable

Now that we have simplified the equation, we can isolate the variable $z$. Adding $20z$ to both sides of the equation gives us:

$10 = 25 + 18z$

Subtracting $25$ from both sides gives us:

$-15 = 18z$

Step 5: Solve for $z$

Finally, we can solve for $z$ by dividing both sides of the equation by $18$. This gives us:

$z = \frac{-15}{18}$

Simplifying further, we get:

$z = -\frac{5}{6}$

Conclusion

In this article, we have solved the equation $\frac{1}{z} - 2 = \frac{5}{2z} - \frac{1}{5}$ for the variable $z$. We started by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, then simplified the equation and isolated the variable. Finally, we solved for $z$ by dividing both sides of the equation by the coefficient of $z$. The solution to the equation is $z = -\frac{5}{6}$.

Tips and Tricks

• When solving equations involving fractions, it is essential to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
• Simplifying the equation as you go can help make the solution process easier and less prone to errors.
• Isolating the variable is a crucial step in solving equations, and it can be achieved by adding or subtracting the same value from both sides of the equation.

Real-World Applications

Solving equations is a fundamental concept in mathematics, and it has numerous real-world applications. In physics, equations are used to describe the motion of objects and the behavior of physical systems. In engineering, equations are used to design and optimize systems, such as bridges and buildings. In economics, equations are used to model economic systems and make predictions about future trends.

Final Thoughts

Solving equations is a critical skill in mathematics, and it requires a deep understanding of algebraic concepts and techniques. By following the steps outlined in this article, you can solve equations involving fractions and variables. Remember to eliminate fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, simplify the equation as you go, and isolate the variable. With practice and patience, you can become proficient in solving equations and apply your skills to real-world problems.

Introduction

Solving equations involving fractions and variables can be a challenging task, but with the right approach and techniques, it can be made easier. In this article, we will provide a Q&A section to help you understand the concepts and techniques involved in solving equations involving fractions and variables.

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

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

Q: How do I find the least common multiple (LCM) of the denominators?

A: 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 all of them. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: What if the equation has multiple fractions with different denominators?

A: If the equation has multiple fractions with different denominators, you need to find the least common multiple (LCM) of all the denominators and multiply both sides of the equation by the LCM.

Q: How do I simplify the equation after eliminating the fractions?

A: After eliminating the fractions, you can simplify the equation by combining like terms and performing any necessary operations to isolate the variable.

Q: What if the equation has a variable in the denominator?

A: If the equation has a variable in the denominator, you need to be careful when multiplying both sides of the equation by the LCM. You may need to use a different approach, such as multiplying both sides of the equation by the reciprocal of the variable.

Q: How do I know if I have solved the equation correctly?

A: To check if you have solved the equation correctly, you can plug the solution back into the original equation and verify that it is true.

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

A: Some common mistakes to avoid when solving equations involving fractions and variables include:

• Not eliminating the fractions before simplifying the equation
• Not isolating the variable correctly
• Not checking the solution by plugging it back into the original equation

Q: How can I practice solving equations involving fractions and variables?

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

Q: What are some real-world applications of solving equations involving fractions and variables?

A: Solving equations involving fractions and variables has numerous real-world applications, including:

• Physics: equations are used to describe the motion of objects and the behavior of physical systems
• Engineering: equations are used to design and optimize systems, such as bridges and buildings
• Economics: equations are used to model economic systems and make predictions about future trends

Conclusion

Solving equations involving fractions and variables can be a challenging task, but with the right approach and techniques, it can be made easier. By following the steps outlined in this article and practicing regularly, you can become proficient in solving equations involving fractions and variables and apply your skills to real-world problems.