Solving Linear Equations: A Step-by-Step Guide with Practical Examples

Linear equations are fundamental to understanding algebra and are widely used in various fields such as physics, engineering, and finance. One of the most basic problems in algebra is solving for the value of (x) in a linear equation. In this article, we will explore the process of solving a linear equation through a practical example and provide tips to help you solve similar problems.

Understanding Linear Equations

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable raised to the first power. The equation takes the form of (ax b c), where (a), (b), and (c) are constants and (x) is the variable. Solving a linear equation means finding the value of (x) that satisfies the equation.

Step-by-Step Guide: Solving (9x - 45 12x - 60)

Let's solve the equation (9x - 45 12x - 60). This is a common type of linear equation where we need to isolate the variable (x). Here are the steps to follow:

1. Combine Like Terms

First, we need to move all terms containing (x) to one side of the equation and the constant terms to the other side. We can do this by subtracting (12x) and adding (45) to both sides of the equation:

[9x - 12x -60 - 45]

This simplifies to:

[-3x -15]

2. Solve for (x)

Next, we divide both sides of the equation by (-3) to isolate (x):

[x frac{-15}{-3} 5]

Therefore, (x 5) is the solution to the given equation.

Understanding the Solution

In the given equation (9x - 45 12x - 60), we found that (x 5). To verify this, we can substitute (x 5) back into the original equation:

For the left-hand side (LHS):

[9(5) - 45 45 - 45 0]

For the right-hand side (RHS):

[12(5) - 60 60 - 60 0]

Since both sides are equal, our solution is confirmed to be correct.

Practical Applications of Linear Equations

Linear equations have numerous practical applications in real-world scenarios. For example:

Finance: Linear equations can be used to model and solve financial problems, such as calculating the break-even point in a business or determining the interest earned on an investment. Physics: Linear equations are used to describe the motion of objects in a straight line, such as the relationship between distance, time, and velocity. Engineering: Linear equations are used in electrical circuits to understand the relationship between voltage, current, and resistance.

Common Mistakes to Avoid

When solving linear equations, there are a few common mistakes to watch out for:

Moving terms without changing sign: When moving terms across the equal sign, remember to change their sign. Forgetting to apply the same operation to both sides: Every step you perform on one side of the equation must also be performed on the other side to maintain the equality. Misinterpreting the order of operations: Always follow the correct order of operations, especially when dealing with negative numbers or fractions.

Conclusion

Solving linear equations is an essential skill in algebra and has widespread applications in various fields. By understanding the process and common mistakes to avoid, you can solve even complex linear equations with confidence. This article has provided a step-by-step guide to solving the equation (9x - 45 12x - 60) and highlighted the importance of accurate and systematic problem-solving in linear equations.

Mastering linear equations opens the door to more advanced mathematical concepts and real-world problem-solving. Whether you are a student or a professional, the ability to solve linear equations is a valuable skill to have in your repertoire.