Solve For { Y $} : : : { (6y) - 5 = 7 \}

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a simple linear equation of the form (6y) - 5 = 7. We will break down the solution step by step, using clear and concise language to ensure that readers understand the process.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it means. The equation (6y) - 5 = 7 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 simple one, with only one variable and a constant on the right-hand side.

Step 1: Add 5 to Both Sides

To solve the equation, we need to isolate the variable y. The first step is to add 5 to both sides of the equation. This will get rid of the negative term on the left-hand side.

(6y) - 5 + 5 = 7 + 5

Simplifying the left-hand side, we get:

6y = 12

Step 2: Divide Both Sides by 6

Now that we have isolated the term with the variable, we can divide both sides of the equation by 6 to solve for y.

6y / 6 = 12 / 6

Simplifying the left-hand side, we get:

y = 2

Conclusion

In this article, we solved a simple linear equation of the form (6y) - 5 = 7. We broke down the solution into two steps: adding 5 to both sides and dividing both sides by 6. By following these steps, we were able to isolate the variable y and find its value.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • When adding or subtracting the same value to both sides of an equation, you can simply cancel out the value.
  • When dividing both sides of an equation by a value, make sure to divide both the left-hand side and the right-hand side by that value.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Common Mistakes

  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
  • Not isolating the variable: Failing to isolate the variable can make it difficult to solve the equation.
  • Not checking the solution: Failing to check the solution can lead to incorrect answers.

Conclusion

Introduction

In our previous article, we covered the basics of solving linear equations, including the step-by-step process for solving an equation of the form (6y) - 5 = 7. In this article, we will answer some common questions that students often have when it comes to 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. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

  • The equation is in the form ax + b = c.
  • The highest power of the variable is 1.
  • There are no squared or cubed variables.

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. For example, the equation 2x + 3 = 5 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

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

A: To solve a linear equation with fractions, follow these steps:

  1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
  2. Simplify the equation.
  3. Solve for the variable.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution?

A: To check your solution, plug the value of the variable back into the original equation and simplify. If the equation is true, then your solution is correct.

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

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

  • Not following the order of operations.
  • Not isolating the variable.
  • Not checking the solution.
  • Not simplifying the equation.

Q: How do I use linear equations in real-world applications?

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

  • Finance: Linear equations are used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Linear equations are used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and avoiding common mistakes, readers can solve linear equations with confidence. Remember to follow the order of operations, isolate the variable, and check the solution to ensure accuracy.