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

Equations form the cornerstone of algebra and mathematics. They allow us to manipulate numerical expressions to find the unknown values. In this article, we will explore how to solve the equation 3x 3 -2x - 12 step-by-step, using a simple analogy to understanding the balance of a set of scales. This method, often referred to as the algebraic method, can be applied to a wide range of equations, making it a valuable tool for students and professionals alike.

Step 1: Combine Like Terms

The first step in solving the equation is to combine like terms. This means we need to get all the x terms on one side of the equation and all the numerical terms on the other. Let's start with the given equation:

3x 3 -2x - 12

To combine like terms, we add 2x to both sides of the equation. This step is akin to adding the same weight to both pans of a balance to keep the scales level:

3x 3 2x -2x - 12 2x

This simplifies to:

5x 3 -12

Step 2: Isolate the x Term

The next step is to isolate the x term. We do this by subtracting 3 from both sides of the equation. This step is like removing the same weight from both pans to keep the balance:

5x 3 - 3 -12 - 3

This simplifies to:

5x -15

Step 3: Solve for x

The final step is to solve for x by dividing both sides of the equation by 5. This step can be explained as dividing both pans of the balance equally:

5x / 5 -15 / 5

This simplifies to:

x -3

Understanding Equations as Balances

At a basic level, you can think of an equation as a set of weighing scales with two pans on either side of a support. If the pans are level, the weights are equal. When working with equations, any action you perform on one side of the equation must be done on the other side to maintain balance. For example:

In the equation 3x 3 -2x - 12:

Adding 2x to both sides: This is like adding the same weight to both pans, keeping the balance level. Subtracting 3 from both sides: This is like removing the same weight from both pans, keeping the balance level. Dividing both sides by 5: This is like dividing both pans equally, finding the value of x.

Practical Example

Let's apply the steps to a practical example:

2x 7 5x - 11

Step 1: Combine Like Terms

Add 2x to both sides:

2x 7 2x 5x - 11 2x

This simplifies to:

4x 7 5x - 11

Step 2: Isolate the x Term

Subtract 4x from both sides:

4x 7 - 4x 5x - 11 - 4x

This simplifies to:

7 x - 11

Step 3: Solve for x

Add 11 to both sides:

7 11 x - 11 11

This simplifies to:

x 18

Conclusion

Understanding and applying the principles of solving equations can be a powerful tool in mathematics. By using the analogy of weighing scales, you can better grasp the concept of maintaining balance when solving equations. This method, known as the algebraic method, is not only effective but also provides a clear visualization of the process.

Remember, practice is key to mastering these skills. With time and consistent effort, solving equations will become second nature.