Solve For \[$ Y \$\] In The Equation:$\[ A \cdot (n + Y) = 10y + 32 \\]

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Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. In this article, we will focus on solving for ${ y }$ in the given equation: ${ a \cdot (n + y) = 10y + 32 }$. This equation involves variables ${ a }$, ${ n }$, and ${ y }$, and our goal is to isolate ${ y }$ and find its value.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand its components. The equation is a linear equation, which means it can be written in the form ${ ax = b }$, where ${ a }$ and ${ b }$ are constants, and ${ x }$ is the variable. In this case, the equation is ${ a \cdot (n + y) = 10y + 32 }$.

Distributive Property

To solve for ${ y }$, we need to isolate it on one side of the equation. We can start by applying the distributive property, which states that ${ a \cdot (b + c) = a \cdot b + a \cdot c }$. Applying this property to the equation, we get:

${ a \cdot n + a \cdot y = 10y + 32 }$

Isolate ${ y }$

Now that we have applied the distributive property, we can isolate ${ y }$ by moving all the terms involving ${ y }$ to one side of the equation. We can do this by subtracting ${ 10y }$ from both sides of the equation:

${ a \cdot n + a \cdot y - 10y = 32 }$

Combine Like Terms

Next, we can combine like terms by adding or subtracting the coefficients of ${ y }$. In this case, we have ${ a \cdot y }$ and ${ -10y }$, which can be combined as:

${ a \cdot y - 10y = (a - 10)y }$

So, the equation becomes:

${ a \cdot n + (a - 10)y = 32 }$

Isolate ${ y }$ Again

Now that we have combined like terms, we can isolate ${ y }$ again by moving all the terms involving ${ y }$ to one side of the equation. We can do this by subtracting ${ a \cdot n }$ from both sides of the equation:

${ (a - 10)y = 32 - a \cdot n }$

Solve for ${ y }$

Finally, we can solve for ${ y }$ by dividing both sides of the equation by ${ (a - 10) }$:

${ y = \frac{32 - a \cdot n}{a - 10} }$

Conclusion

In this article, we have solved for ${ y }$ in the equation ${ a \cdot (n + y) = 10y + 32 }$. We applied the distributive property, isolated ${ y }$, combined like terms, and finally solved for ${ y }$. The solution is ${ y = \frac{32 - a \cdot n}{a - 10} }$.

Example

Let's consider an example to illustrate the solution. Suppose we have ${ a = 2 }$, ${ n = 3 }$, and we want to find the value of ${ y }$. Plugging these values into the solution, we get:

${ y = \frac{32 - 2 \cdot 3}{2 - 10} }$

${ y = \frac{32 - 6}{-8} }$

${ y = \frac{26}{-8} }$

${ y = -\frac{13}{4} }$

Therefore, the value of ${ y }$ is ${ -\frac{13}{4} }$.

Applications

Solving equations like this one has many applications in mathematics and real-world problems. For example, in physics, we may need to solve equations to find the position or velocity of an object. In economics, we may need to solve equations to find the equilibrium price or quantity of a good. In computer science, we may need to solve equations to find the solution to a system of linear equations.

Final Thoughts

In conclusion, solving for ${ y }$ in the equation ${ a \cdot (n + y) = 10y + 32 }$ requires careful application of mathematical concepts and techniques. By following the steps outlined in this article, we can isolate ${ y }$ and find its value. The solution is ${ y = \frac{32 - a \cdot n}{a - 10} }$. We hope this article has provided a clear and concise explanation of how to solve this type of equation.

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Introduction

In our previous article, we solved for ${ y }$ in the equation ${ a \cdot (n + y) = 10y + 32 }$. In this article, we will answer some frequently asked questions (FAQs) related to solving this type of equation.

Q: What is the first step in solving the equation ${ a \cdot (n + y) = 10y + 32 }$?

A: The first step in solving the equation is to apply the distributive property, which states that ${ a \cdot (b + c) = a \cdot b + a \cdot c }$. This will help us to expand the left-hand side of the equation and make it easier to isolate ${ y }$.

Q: How do I isolate ${ y }$ in the equation?

A: To isolate ${ y }$, we need to move all the terms involving ${ y }$ to one side of the equation. We can do this by subtracting ${ 10y }$ from both sides of the equation, and then combining like terms.

Q: What is the difference between ${ a \cdot y }$ and ${ 10y }$?

A: ${ a \cdot y }$ is a term that involves the variable ${ y }$ and the constant ${ a }$, while ${ 10y }$ is a term that involves only the variable ${ y }$. When we combine these two terms, we get ${ (a - 10)y }$.

Q: How do I solve for ${ y }$ in the equation?

A: To solve for ${ y }$, we need to divide both sides of the equation by ${ (a - 10) }$. This will give us the value of ${ y }$ in terms of ${ a }$ and ${ n }$.

Q: What is the final solution to the equation?

A: The final solution to the equation is ${ y = \frac{32 - a \cdot n}{a - 10} }$.

Q: Can I use this solution to solve for ${ y }$ in other equations?

A: Yes, you can use this solution to solve for ${ y }$ in other equations that have a similar form. However, you will need to adjust the solution to fit the specific equation you are working with.

Q: What are some common mistakes to avoid when solving this type of equation?

A: Some common mistakes to avoid when solving this type of equation include:

• Not applying the distributive property correctly
• Not isolating ${ y }$ correctly
• Not combining like terms correctly
• Not solving for ${ y }$ correctly

Q: How can I practice solving this type of equation?

A: You can practice solving this type of equation by working through examples and exercises. You can also try solving different types of equations to see how the solution changes.

Q: What are some real-world applications of solving this type of equation?

A: Solving this type of equation has many real-world applications, including:

• Physics: solving equations to find the position or velocity of an object
• Economics: solving equations to find the equilibrium price or quantity of a good
• Computer science: solving equations to find the solution to a system of linear equations

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to solving the equation ${ a \cdot (n + y) = 10y + 32 }$. We hope this article has provided a clear and concise explanation of how to solve this type of equation, and has helped to address some common questions and concerns.