Solve For \[$ Y \$\]:$\[ 3 + 6y = 15 \\]

Introduction

In mathematics, solving for a variable is a fundamental concept that is used to find the value of a variable in an equation. In this article, we will focus on solving for the variable $y$ in the equation $3 + 6y = 15$. This equation is a linear equation, and solving for $y$ will involve isolating the variable on one side of the equation.

Understanding the Equation

The given equation is $3 + 6y = 15$. This equation is a linear equation because it is in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. In this equation, $a = 6$, $b = 3$, and $c = 15$. The variable $y$ is the unknown quantity that we need to solve for.

Isolating the Variable

To solve for $y$, we need to isolate the variable on one side of the equation. This can be done by subtracting $3$ from both sides of the equation. This will give us $6y = 15 - 3$, which simplifies to $6y = 12$.

Solving for $y$

Now that we have isolated the variable, we can solve for $y$ by dividing both sides of the equation by $6$. This will give us $y = \frac{12}{6}$, which simplifies to $y = 2$.

Conclusion

In this article, we solved for the variable $y$ in the equation $3 + 6y = 15$. We started by understanding the equation and isolating the variable on one side of the equation. Then, we solved for $y$ by dividing both sides of the equation by $6$. The final answer is $y = 2$.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

1. Understand the equation: The given equation is $3 + 6y = 15$.
2. Isolate the variable: Subtract $3$ from both sides of the equation to get $6y = 15 - 3$, which simplifies to $6y = 12$.
3. Solve for $y$: Divide both sides of the equation by $6$ to get $y = \frac{12}{6}$, which simplifies to $y = 2$.

Tips and Tricks

Here are some tips and tricks to help you solve for variables in linear equations:

• Understand the equation: Before you start solving for the variable, make sure you understand the equation and what it means.
• Isolate the variable: To solve for the variable, you need to isolate it on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation.
• Use inverse operations: To solve for the variable, you need to use inverse operations to isolate the variable. For example, if you have $6y = 12$, you can divide both sides of the equation by $6$ to get $y = \frac{12}{6}$.

Real-World Applications

Solving for variables in linear equations has many real-world applications. Here are a few examples:

• Finance: In finance, solving for variables in linear equations can help you calculate interest rates, investment returns, and other financial metrics.
• Science: In science, solving for variables in linear equations can help you calculate rates of change, acceleration, and other scientific metrics.
• Engineering: In engineering, solving for variables in linear equations can help you design and optimize systems, such as bridges, buildings, and other structures.

Conclusion

In conclusion, solving for variables in linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the equation, isolating the variable, and using inverse operations, you can solve for variables in linear equations. With practice and experience, you can become proficient in solving for variables in linear equations and apply this skill to a wide range of real-world problems.

Frequently Asked Questions

Here are some frequently asked questions about solving for variables in linear equations:

• Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable is $1$.
• Q: How do I isolate the variable in a linear equation? A: To isolate the variable, you need to use inverse operations to get the variable on one side of the equation.
• Q: What are some real-world applications of solving for variables in linear equations? A: Solving for variables in linear equations has many real-world applications, including finance, science, and engineering.

Final Answer

The final answer is $y = 2$.

Introduction

In our previous article, we solved for the variable $y$ in the equation $3 + 6y = 15$. In this article, we will provide a Q&A section to help you understand the concept of solving for variables in linear equations.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is $1$. For example, $2x + 3 = 5$ is a linear equation because the highest power of the variable $x$ is $1$.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable, you need to use inverse operations to get the variable on one side of the equation. For example, if you have $2x + 3 = 5$, you can subtract $3$ from both sides of the equation to get $2x = 5 - 3$, which simplifies to $2x = 2$. Then, you can divide both sides of the equation by $2$ to get $x = \frac{2}{2}$, which simplifies to $x = 1$.

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

A: Some common mistakes to avoid when solving for variables in linear equations include:

• Not isolating the variable: Make sure to isolate the variable on one side of the equation.
• Not using inverse operations: Use inverse operations to get the variable on one side of the equation.
• Not checking your work: Check your work to make sure that you have solved for the variable correctly.

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

A: To check your work, you can plug your solution back into the original equation to make sure that it is true. For example, if you have solved for $x$ in the equation $2x + 3 = 5$, you can plug $x = 1$ back into the equation to get $2(1) + 3 = 5$, which simplifies to $5 = 5$. This shows that your solution is correct.

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

A: Solving for variables in linear equations has many real-world applications, including finance, science, and engineering. For example, in finance, you can use linear equations to calculate interest rates and investment returns. In science, you can use linear equations to calculate rates of change and acceleration. In engineering, you can use linear equations to design and optimize systems.

Q: How do I solve for variables in linear equations with fractions?

A: To solve for variables in linear equations with fractions, you can use the same steps as you would for solving for variables in linear equations with whole numbers. For example, if you have the equation $\frac{2}{3}x + 1 = 2$, you can subtract $1$ from both sides of the equation to get $\frac{2}{3}x = 2 - 1$, which simplifies to $\frac{2}{3}x = 1$. Then, you can multiply both sides of the equation by $\frac{3}{2}$ to get $x = \frac{3}{2} \cdot 1$, which simplifies to $x = \frac{3}{2}$.

Q: How do I solve for variables in linear equations with decimals?

A: To solve for variables in linear equations with decimals, you can use the same steps as you would for solving for variables in linear equations with whole numbers. For example, if you have the equation $2.5x + 3 = 5$, you can subtract $3$ from both sides of the equation to get $2.5x = 5 - 3$, which simplifies to $2.5x = 2$. Then, you can divide both sides of the equation by $2.5$ to get $x = \frac{2}{2.5}$, which simplifies to $x = 0.8$.

Conclusion

In conclusion, solving for variables in linear equations is a fundamental concept in mathematics that has many real-world applications. By understanding the equation, isolating the variable, and using inverse operations, you can solve for variables in linear equations. With practice and experience, you can become proficient in solving for variables in linear equations and apply this skill to a wide range of real-world problems.

Final Answer

The final answer is $y = 2$.