Solving Linear Equations: A Comprehensive Guide with a Focus on 5y - 2y -24

In the field of mathematics, understanding and solving linear equations is a fundamental skill. This article will guide you through the process of solving a specific linear equation, 5y - 2y -24, and also provide a broader understanding of linear equations in general. We will cover the step-by-step solution process, the importance of linear equations, and useful tips for tackling similar equations.

Introduction to Linear Equations

Linear equations are algebraic expressions where the variables, such as y in this case, appear only to the first power. These equations form straight lines when graphed, hence the term 'linear.' They are widely used in various fields, including physics, economics, and engineering, due to their simplicity and effectiveness in modeling real-world scenarios.

Understanding the Equation: 5y - 2y -24

Let's start by understanding the given equation:

5y - 2y -24

This equation involves a variable, y, on the left side, and a constant, -24, on the right side. The process of solving it involves isolating the variable, which makes it possible to find the value of y.

Step-by-Step Solution

To solve the equation, we need to perform algebraic operations to isolate the variable y.

Combine Like Terms: The first step is to combine the like terms on the left side of the equation. Simplify the Equation: After combining like terms, the equation simplifies to: Isolate the Variable: Once the equation is simplified, isolate y by performing algebraic operations. Solve for y: Finally, solve for y by dividing both sides of the equation by the coefficient of y.

Breaking Down Each Step

Step 1: Combine Like Terms
5y - 2y -24

Combine the like terms on the left side of the equation:

(5y - 2y) -24

7y -24

Step 2: Simplify the Equation
As we have already combined the like terms, the equation is now:

7y -24

Step 3: Isolate the Variable
To isolate y, we divide both sides of the equation by 7:

y frac{-24}{7}

Step 4: Solve for y
Finally, solve for y:

y -frac{24}{7} ≈ -3.4285

Importance of Linear Equations

Linear equations are not only fundamental in algebra but also have practical applications in various real-world scenarios. They are used in:

Physics: To model motion, energy conservation, and many other physical phenomena. Economics: To analyze supply and demand, cost functions, and market trends. Engineering: To solve problems related to structural integrity, fluid dynamics, and electrical circuits.

Additional Resources and Tips

Here are some additional resources and tips for solving linear equations:

Practice Problems: Regularly practice solving different types of linear equations to improve your understanding and skills. Graphing: Use graphing techniques to visualize the solutions and understand the behavior of the equation. Online Tools: Utilize online calculators and solvers for quick and accurate solutions.

Conclusion

By following the step-by-step process, we can solve linear equations like 5y - 2y -24. Understanding linear equations is crucial for many practical applications and forms the basis for more advanced mathematical concepts. Whether you are a student, a professional, or simply interested in mathematics, mastering the art of solving linear equations is a valuable skill.