Solve The Equation For \[$ Y \$\]:$\[ Y + 9 = \frac{5}{2}(x + 6) \\]

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable y. We will use the given equation as an example and provide a step-by-step guide on how to solve it.

The Given Equation

The given equation is:

y + 9 = (5/2)(x + 6)

Understanding the Equation

Before we start solving the equation, let's break it down and understand what it means. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, y) is 1. The equation is also a quadratic equation in disguise, as it involves a quadratic expression on the right-hand side.

Step 1: Isolate the Variable y

To solve the equation, we need to isolate the variable y. This means we need to get y by itself on one side of the equation. To do this, we can start by subtracting 9 from both sides of the equation.

y + 9 - 9 = (5/2)(x + 6) - 9

This simplifies to:

y = (5/2)(x + 6) - 9

Step 2: Simplify the Right-Hand Side

Now that we have isolated the variable y, we can simplify the right-hand side of the equation. To do this, we can start by distributing the 5/2 to the terms inside the parentheses.

y = (5/2)x + (5/2)(6) - 9

This simplifies to:

y = (5/2)x + 15 - 9

Step 3: Combine Like Terms

Now that we have simplified the right-hand side, we can combine like terms. In this case, we have a constant term of 15 and a constant term of -9. We can combine these terms by adding them together.

y = (5/2)x + 6

Conclusion

In this article, we have solved a linear equation by isolating the variable y. We started by subtracting 9 from both sides of the equation, then simplified the right-hand side by distributing the 5/2 to the terms inside the parentheses. Finally, we combined like terms to get the final solution.

Tips and Tricks

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

  • Always start by isolating the variable you are trying to solve for.
  • Use inverse operations to get rid of any constants or coefficients that are attached to the variable.
  • Simplify the right-hand side of the equation by distributing any coefficients to the terms inside the parentheses.
  • Combine like terms to get the final solution.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

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

  • Not isolating the variable you are trying to solve for.
  • Not using inverse operations to get rid of any constants or coefficients that are attached to the variable.
  • Not simplifying the right-hand side of the equation by distributing any coefficients to the terms inside the parentheses.
  • Not combining like terms to get the final solution.

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations by isolating the variable y. However, we know that practice makes perfect, and the best way to learn is by doing. In this article, we will provide a Q&A guide to help you practice solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, y) is 1. It is a simple equation that can be solved by using basic algebraic operations.

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

A: To isolate the variable y, you need to get y by itself on one side of the equation. This can be done by using inverse operations, such as addition, subtraction, multiplication, and division.

Q: What is an inverse operation?

A: An inverse operation is an operation that undoes another operation. For example, addition and subtraction are inverse operations, as are multiplication and division.

Q: How do I simplify the right-hand side of a linear equation?

A: To simplify the right-hand side of a linear equation, you need to distribute any coefficients to the terms inside the parentheses. This will help you to combine like terms and get the final solution.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms, as are 3y and 5y.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have 2x + 4x, you can combine them by adding the coefficients: 2x + 4x = 6x.

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

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

  • Not isolating the variable you are trying to solve for.
  • Not using inverse operations to get rid of any constants or coefficients that are attached to the variable.
  • Not simplifying the right-hand side of the equation by distributing any coefficients to the terms inside the parentheses.
  • Not combining like terms to get the final solution.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Using online resources, such as Khan Academy or Mathway.
  • Working with a tutor or teacher to get personalized help.
  • Practicing with worksheets or exercises.
  • Solving real-world problems that involve linear equations.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and practicing with real-world problems, you can become proficient in solving linear equations. Remember to always isolate the variable you are trying to solve for, use inverse operations to get rid of any constants or coefficients that are attached to the variable, simplify the right-hand side of the equation by distributing any coefficients to the terms inside the parentheses, and combine like terms to get the final solution.