Solving The Linear Equation: 3x - 2 = -4x - 65

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Solving linear equations is a fundamental skill in algebra, and understanding the process is crucial for tackling more complex mathematical problems. In this comprehensive guide, we will walk you through the step-by-step solution of the equation 3x - 2 = -4x - 65. We'll break down each step, explain the underlying principles, and provide insights to help you master this essential algebraic technique. By the end of this article, you'll not only know how to solve this specific equation but also have a solid foundation for solving other linear equations.

Understanding Linear Equations

Before diving into the solution, it's important to understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because when plotted on a graph, they form a straight line. The general form of a linear equation in one variable is ax + b = c, where x is the variable, and a, b, and c are constants. Recognizing this form is the first step in solving any linear equation.

In our case, the equation 3x - 2 = -4x - 65 fits this form, making it a linear equation. The goal is to isolate the variable x on one side of the equation to find its value. This involves performing algebraic operations on both sides of the equation while maintaining balance. Let's explore the step-by-step process to achieve this.

Step 1: Combine Like Terms

The first step in solving 3x - 2 = -4x - 65 is to combine like terms. Like terms are those that contain the same variable raised to the same power (in this case, x) or are constants. To combine like terms, we need to move all terms containing x to one side of the equation and all constant terms to the other side. This is done by adding or subtracting terms from both sides of the equation. Remember, whatever operation you perform on one side, you must also perform on the other side to maintain equality.

To move the -4x term from the right side to the left side, we add 4x to both sides of the equation:

3x - 2 + 4x = -4x - 65 + 4x

This simplifies to:

7x - 2 = -65

Now, we need to move the constant term -2 from the left side to the right side. To do this, we add 2 to both sides of the equation:

7x - 2 + 2 = -65 + 2

This simplifies to:

7x = -63

By performing these operations, we have successfully combined like terms and simplified the equation. The next step involves isolating the variable x.

Step 2: Isolate the Variable

After combining like terms, the equation is now in the form 7x = -63. The next step is to isolate the variable x. This means getting x by itself on one side of the equation. Since x is being multiplied by 7, we need to perform the inverse operation, which is division. We divide both sides of the equation by 7 to isolate x:

(7x) / 7 = (-63) / 7

This simplifies to:

x = -9

Thus, we have isolated the variable x and found its value. This completes the solution of the equation. However, it's always a good practice to verify the solution to ensure accuracy.

Step 3: Verify the Solution

To verify the solution, we substitute the value of x back into the original equation 3x - 2 = -4x - 65. If the left side of the equation equals the right side, then the solution is correct. Substituting x = -9 into the equation, we get:

3(-9) - 2 = -4(-9) - 65

Now, we simplify both sides:

-27 - 2 = 36 - 65

-29 = -29

Since both sides of the equation are equal, our solution x = -9 is correct. This verification step confirms that we have solved the equation accurately.

Alternative Methods for Solving Linear Equations

While the method described above is a standard approach, there are alternative methods for solving linear equations that can be useful in different situations. One such method involves using the distributive property to simplify equations with parentheses or fractions. Another approach is to use a graphical method, where the equation is represented as a line on a graph, and the solution is found by identifying the point where the line intersects the x-axis. Understanding these alternative methods can provide additional flexibility and insight into solving linear equations.

Common Mistakes to Avoid

When solving linear equations, it's important to avoid common mistakes that can lead to incorrect solutions. One frequent error is not performing the same operation on both sides of the equation, which violates the principle of equality. Another mistake is incorrectly applying the order of operations (PEMDAS/BODMAS) when simplifying expressions. Additionally, errors can occur when combining like terms or distributing values across parentheses. By being mindful of these potential pitfalls, you can improve your accuracy and confidence in solving linear equations.

Real-World Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including physics, engineering, economics, and computer science. For example, linear equations can model the relationship between distance, speed, and time, or the relationship between supply and demand in economics. Understanding how to solve linear equations can be valuable in solving practical problems and making informed decisions in various contexts.

Practice Problems

To solidify your understanding of solving linear equations, it's essential to practice with a variety of problems. Here are a few practice problems you can try:

  1. Solve for x: 5x + 3 = 2x - 9
  2. Solve for y: -2y - 7 = 3y + 8
  3. Solve for z: 4(z - 2) = -2(z + 5)

Working through these problems will help you reinforce the concepts and techniques discussed in this article. Remember to follow the steps outlined earlier: combine like terms, isolate the variable, and verify your solution.

Conclusion

In this comprehensive guide, we have explored the step-by-step solution of the linear equation 3x - 2 = -4x - 65. We discussed the importance of understanding linear equations, the process of combining like terms, isolating the variable, and verifying the solution. We also touched on alternative methods for solving linear equations, common mistakes to avoid, and real-world applications of these equations. By mastering the techniques presented here, you will be well-equipped to solve a wide range of linear equations and apply these skills in various contexts.

To further enhance your understanding of linear equations and related topics, you may find the resources available at Khan Academy's Algebra I section to be quite beneficial. This website offers a wide array of lessons, practice exercises, and quizzes that can help you solidify your grasp of these crucial algebraic concepts.