Solve For $v$.$\[ \frac{1}{2} V + 8 = 3 - \frac{1}{3} V \\]$v = \square$

Solving for $v$: A Step-by-Step Guide to Isolating the Variable

In mathematics, solving for a variable is a fundamental concept that involves isolating the variable on one side of the equation. This is a crucial skill that is used extensively in various mathematical operations, including algebra, geometry, and calculus. In this article, we will focus on solving for the variable $v$ in the given equation.

The Given Equation

The given equation is:

$\frac{1}{2} v + 8 = 3 - \frac{1}{3} v$

Our goal is to isolate the variable $v$ on one side of the equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by getting rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 6.

# Import necessary modules
import sympy as sp

# Define the variable
v = sp.symbols('v')

# Define the equation
eq = sp.Eq((1/2)*v + 8, 3 - (1/3)*v)

# Multiply both sides of the equation by 6
eq = sp.Eq(6*((1/2)*v + 8), 6*(3 - (1/3)*v))

This simplifies the equation to:

$3v + 48 = 18 - 2v$

Step 2: Combine Like Terms

Next, we can combine like terms on both sides of the equation.

# Combine like terms on both sides of the equation
eq = sp.Eq(3*v + 48, 18 - 2*v)

This simplifies the equation to:

$5v + 48 = 18$

Step 3: Isolate the Variable

Now, we can isolate the variable $v$ by subtracting 48 from both sides of the equation.

# Subtract 48 from both sides of the equation
eq = sp.Eq(5*v + 48 - 48, 18 - 48)

This simplifies the equation to:

$5v = -30$

Step 4: Solve for $v$

Finally, we can solve for $v$ by dividing both sides of the equation by 5.

# Divide both sides of the equation by 5
eq = sp.Eq((5*v)/5, (-30)/5)

This simplifies the equation to:

$v = -6$

In this article, we have solved for the variable $v$ in the given equation. We started by simplifying the equation, combining like terms, isolating the variable, and finally solving for $v$. The final solution is $v = -6$. This demonstrates the importance of following the order of operations and using algebraic techniques to solve for variables in mathematical equations.

Example Use Cases

Solving for variables is a crucial skill that is used extensively in various mathematical operations. Here are some example use cases:

• Algebra: Solving for variables is a fundamental concept in algebra. It is used to solve linear equations, quadratic equations, and other types of equations.
• Geometry: Solving for variables is used to solve geometric problems, such as finding the length of a side of a triangle or the area of a circle.
• Calculus: Solving for variables is used to solve optimization problems, such as finding the maximum or minimum value of a function.

Tips and Tricks

Here are some tips and tricks for solving for variables:

• Follow the order of operations: When solving for variables, it is essential to follow the order of operations (PEMDAS).
• Use algebraic techniques: Algebraic techniques, such as combining like terms and isolating the variable, are essential for solving for variables.
• Check your work: It is essential to check your work to ensure that the solution is correct.

By following these tips and tricks, you can become proficient in solving for variables and apply this skill to various mathematical operations.
Solving for $v$: A Q&A Guide to Isolating the Variable

In our previous article, we solved for the variable $v$ in the given equation. However, we understand that solving for variables can be a challenging task, especially for those who are new to algebra. In this article, we will provide a Q&A guide to help you understand the concept of solving for variables and how to apply it to various mathematical operations.

Q: What is solving for variables?

A: Solving for variables is a fundamental concept in mathematics that involves isolating the variable on one side of the equation. This is a crucial skill that is used extensively in various mathematical operations, including algebra, geometry, and calculus.

Q: Why is solving for variables important?

A: Solving for variables is important because it allows you to find the value of the variable, which is essential in solving mathematical problems. By isolating the variable, you can determine the value of the variable and use it to solve other mathematical problems.

Q: How do I solve for variables?

A: To solve for variables, you need to follow the order of operations (PEMDAS) and use algebraic techniques, such as combining like terms and isolating the variable. Here are the steps to follow:

1. Simplify the equation: Start by simplifying the equation by getting rid of any fractions or decimals.
2. Combine like terms: Combine like terms on both sides of the equation to simplify it further.
3. Isolate the variable: Isolate the variable by adding or subtracting the same value from both sides of the equation.
4. Solve for the variable: Finally, solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: What are some common mistakes to avoid when solving for variables?

A: Here are some common mistakes to avoid when solving for variables:

• Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
• Not combining like terms: Failing to combine like terms can make the equation more complicated and difficult to solve.
• Not isolating the variable: Failing to isolate the variable can make it difficult to solve for the variable.
• Not checking your work: Failing to check your work can lead to incorrect solutions.

Q: How do I check my work when solving for variables?

A: To check your work when solving for variables, follow these steps:

1. Plug in the solution: Plug in the solution into the original equation to see if it is true.
2. Check the equation: Check the equation to make sure that it is balanced and that the variable is isolated.
3. Verify the solution: Verify the solution by checking if it satisfies the original equation.

Q: What are some real-world applications of solving for variables?

A: Solving for variables has many real-world applications, including:

• Science: Solving for variables is used in scientific experiments to determine the value of unknown variables.
• Engineering: Solving for variables is used in engineering to design and optimize systems.
• Economics: Solving for variables is used in economics to model and analyze economic systems.

In this article, we have provided a Q&A guide to help you understand the concept of solving for variables and how to apply it to various mathematical operations. By following the steps outlined in this article, you can become proficient in solving for variables and apply this skill to real-world problems.

Example Use Cases

Here are some example use cases for solving for variables:

• Science: A scientist wants to determine the value of a unknown variable in a scientific experiment. By solving for the variable, the scientist can determine the value of the variable and use it to analyze the results of the experiment.
• Engineering: An engineer wants to design and optimize a system. By solving for the variables, the engineer can determine the value of the variables and use it to design and optimize the system.
• Economics: An economist wants to model and analyze an economic system. By solving for the variables, the economist can determine the value of the variables and use it to model and analyze the economic system.

Tips and Tricks

Here are some tips and tricks for solving for variables:

• Practice, practice, practice: The more you practice solving for variables, the more proficient you will become.
• Use algebraic techniques: Algebraic techniques, such as combining like terms and isolating the variable, are essential for solving for variables.
• Check your work: It is essential to check your work to ensure that the solution is correct.
• Use real-world examples: Using real-world examples can help you understand the concept of solving for variables and how to apply it to real-world problems.