Solve: $y + 9 = 6$y = $

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Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a linear equation. In this article, we will focus on solving the linear equation $y + 9 = 6$ to find the value of $y$.

Understanding the Equation

The given equation is $y + 9 = 6$. This is a linear equation in one variable, where $y$ is the variable. The equation states that the sum of $y$ and $9$ is equal to $6$. Our goal is to isolate the variable $y$ and find its value.

Isolating the Variable

To isolate the variable $y$, we need to get rid of the constant term $9$ on the left-hand side of the equation. We can do this by subtracting $9$ from both sides of the equation. This will give us the value of $y$.

Step-by-Step Solution

Here are the step-by-step steps to solve the equation:

1. Subtract 9 from both sides: $y + 9 - 9 = 6 - 9$
2. Simplify the equation: $y = -3$

Conclusion

In this article, we solved the linear equation $y + 9 = 6$ to find the value of $y$. By isolating the variable $y$ and subtracting $9$ from both sides of the equation, we found that $y = -3$. This is a simple example of solving a linear equation, and it demonstrates the importance of understanding the concept of isolating variables in mathematics.

Real-World Applications

Solving linear equations has numerous real-world applications in various fields, including science, engineering, economics, and finance. For example, in physics, linear equations are used to describe the motion of objects, while in economics, they are used to model the behavior of markets and economies.

Tips and Tricks

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

• Use inverse operations: To isolate the variable, use inverse operations such as addition, subtraction, multiplication, and division.
• Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary variables.
• Check your solution: Check your solution by plugging it back into the original equation to ensure that it is true.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

• Not isolating the variable: Failing to isolate the variable can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can lead to unnecessary complexity.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a linear equation. By following the step-by-step solution and using inverse operations, simplifying the equation, and checking the solution, you can solve linear equations with confidence. Remember to avoid common mistakes such as not isolating the variable, not simplifying the equation, and not checking the solution.

Final Answer

The final answer is: $\boxed{-3}$

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Introduction to Solving Linear Equations Q&A

In our previous article, we solved the linear equation $y + 9 = 6$ to find the value of $y$. In this article, we will provide a Q&A section to help you understand the concept of solving linear equations and provide additional examples and explanations.

Q&A Section

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation in which the variable is not raised to a power greater than 1.

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

A: To isolate the variable, you need to get rid of the constant term on the left-hand side of the equation. You can do this by using inverse operations such as addition, subtraction, multiplication, and division.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that is common to two or more numbers.

Q: How do I solve a linear equation with decimals?

A: To solve a linear equation with decimals, you need to eliminate the decimals by multiplying both sides of the equation by a power of 10.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation in one variable, while a system of linear equations is a set of two or more equations in two or more variables.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to use methods such as substitution, elimination, or graphing to find the values of the variables.

Examples and Explanations

Example 1: Solving a Linear Equation with Fractions

Solve the equation: $\frac{y}{2} + 3 = 5$

Solution:

1. Multiply both sides of the equation by 2 to eliminate the fraction: $y + 6 = 10$
2. Subtract 6 from both sides of the equation to isolate the variable: $y = 4$

Example 2: Solving a Linear Equation with Decimals

Solve the equation: $2.5y + 3 = 7$

Solution:

1. Multiply both sides of the equation by 10 to eliminate the decimal: $25y + 30 = 70$
2. Subtract 30 from both sides of the equation to isolate the variable: $25y = 40$
3. Divide both sides of the equation by 25 to find the value of y: $y = \frac{40}{25}$

Example 3: Solving a System of Linear Equations

Solve the system of equations:

$2x + 3y = 7$ $x - 2y = -3$

Solution:

1. Multiply the second equation by 2 to make the coefficients of y in both equations equal: $2x - 4y = -6$
2. Add the two equations to eliminate the variable y: $4x - 7y = 1$
3. Solve for x: $x = \frac{1 + 7y}{4}$
4. Substitute the value of x into one of the original equations to solve for y: $2(\frac{1 + 7y}{4}) + 3y = 7$

Conclusion

In this article, we provided a Q&A section to help you understand the concept of solving linear equations and provide additional examples and explanations. We also discussed how to solve linear equations with fractions, decimals, and systems of linear equations. Remember to practice solving linear equations to become proficient in this skill.

Final Answer

The final answer is: $\boxed{-3}$