Solving 2(x+3) = 7x - 9 A Step-by-Step Guide

In this article, we will delve into the process of solving a linear equation. Linear equations, forming the bedrock of algebra, are mathematical statements where the highest power of the variable is one. Mastering the techniques to solve these equations is crucial for building a strong foundation in mathematics. We will dissect the equation 2(x + 3) = 7x - 9, providing a comprehensive, step-by-step solution that is easy to grasp, regardless of your mathematical background. Whether you're a student grappling with algebra or someone seeking to refresh their math skills, this guide will equip you with the knowledge and confidence to tackle similar problems. Let's embark on this journey to unravel the mysteries of linear equations, enhancing your problem-solving prowess and deepening your understanding of mathematical principles.

Understanding the Basics of Linear Equations

Before we dive into the specifics of solving 2(x + 3) = 7x - 9, let's first establish a clear understanding of what linear equations are and the fundamental principles that govern their solutions. A linear equation, at its core, is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable, often represented by 'x', is raised to the power of one, hence the term 'linear.' These equations, when graphed on a coordinate plane, produce a straight line, further solidifying their linear nature.

The goal in solving a linear equation is to isolate the variable on one side of the equation, effectively determining the value of the variable that satisfies the equation. This involves manipulating the equation through a series of valid algebraic operations while maintaining the equality. The golden rule of equation solving is that whatever operation you perform on one side of the equation, you must perform the same operation on the other side. This ensures that the equation remains balanced and the solution remains valid.

Some of the key operations used in solving linear equations include:

• Addition and Subtraction: Adding or subtracting the same constant or term to both sides of the equation.
• Multiplication and Division: Multiplying or dividing both sides of the equation by the same non-zero constant.
• Distribution: Applying the distributive property to remove parentheses and simplify expressions.
• Combining Like Terms: Grouping and combining terms with the same variable or constant terms on each side of the equation.

By strategically applying these operations, we can systematically simplify the equation until the variable is isolated, revealing the solution. This step-by-step approach is crucial for accuracy and understanding, particularly when dealing with more complex linear equations. In the following sections, we will apply these principles to solve the equation 2(x + 3) = 7x - 9, illustrating the practical application of these fundamental concepts.

Step-by-Step Solution for 2(x + 3) = 7x - 9

Now, let's embark on the journey of solving the equation 2(x + 3) = 7x - 9 step-by-step. We'll break down each operation and the reasoning behind it to ensure clarity and understanding.

Step 1: Distribute the Constant

The first step involves simplifying the equation by removing the parentheses. We achieve this by applying the distributive property. This property states that a(b + c) = ab + ac. In our equation, we need to distribute the constant 2 across the terms inside the parentheses (x + 3). This means we multiply 2 by both x and 3:

2 * x = 2x

2 * 3 = 6

Applying the distributive property, we rewrite the equation as:

2x + 6 = 7x - 9

This step effectively eliminates the parentheses, making the equation easier to manipulate.

Step 2: Group Like Terms

The next crucial step is to group like terms on either side of the equation. Like terms are terms that contain the same variable raised to the same power or are constants. In our equation, we have 'x' terms (2x and 7x) and constant terms (6 and -9). To isolate the variable, we want to bring all the 'x' terms to one side and all the constant terms to the other side. A common approach is to move the smaller 'x' term to the side with the larger 'x' term to avoid dealing with negative coefficients.

In this case, we have 2x on the left side and 7x on the right side. Since 2x is smaller than 7x, we will subtract 2x from both sides of the equation. This ensures that the 'x' terms are grouped on the right side:

2x + 6 - 2x = 7x - 9 - 2x

Simplifying, we get:

6 = 5x - 9

Now, we need to move the constant terms to the left side. To do this, we add 9 to both sides of the equation:

6 + 9 = 5x - 9 + 9

Simplifying further, we have:

15 = 5x

This step strategically positions the 'x' term and the constant terms on opposite sides, bringing us closer to isolating the variable.

Step 3: Isolate the Variable

The final step in solving for x is to isolate it completely on one side of the equation. Currently, we have 15 = 5x. This means that 5 is multiplying x. To isolate x, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 5:

15 / 5 = 5x / 5

Performing the division, we get:

3 = x

Therefore, the solution to the equation 2(x + 3) = 7x - 9 is:

x = 3

By following these steps – distributing, grouping like terms, and isolating the variable – we have successfully solved the linear equation. This systematic approach can be applied to a wide range of linear equations, building your confidence and proficiency in algebra.

Verifying the Solution

An essential practice in solving equations is verifying the solution. This step ensures that the value we obtained for the variable, in this case, x = 3, truly satisfies the original equation. To verify the solution, we substitute the value of x back into the original equation and check if both sides of the equation are equal.

Our original equation is:

2(x + 3) = 7x - 9

We substitute x = 3 into the equation:

2(3 + 3) = 7(3) - 9

Now, we simplify both sides of the equation following the order of operations (PEMDAS/BODMAS).

Left Side:

2(3 + 3) = 2(6) = 12

Right Side:

7(3) - 9 = 21 - 9 = 12

Comparing both sides, we see that:

12 = 12

Since both sides of the equation are equal when x = 3, our solution is verified. This process of verification is crucial as it confirms the accuracy of our calculations and ensures that the solution is correct. It also helps in identifying any potential errors made during the solving process. By consistently verifying solutions, we reinforce our understanding of the equation-solving process and build confidence in our mathematical abilities.

Common Mistakes to Avoid

Solving linear equations is a fundamental skill in algebra, but it's also an area where common mistakes can occur. Being aware of these pitfalls can help you avoid errors and improve your accuracy. Let's explore some of the frequent mistakes made when solving linear equations and how to prevent them.

1. Incorrect Distribution: A common mistake arises during the distribution step. When multiplying a constant by an expression in parentheses, it's crucial to distribute the constant to every term inside the parentheses. For example, in the equation 2(x + 3) = 7x - 9, forgetting to multiply the 2 by both x and 3 would lead to an incorrect simplification. To avoid this, double-check that you've multiplied the constant by each term inside the parentheses.

2. Sign Errors: Sign errors are another frequent source of mistakes. This often occurs when adding or subtracting terms on both sides of the equation. For instance, when moving a term from one side to the other, remember to change its sign. A positive term becomes negative, and a negative term becomes positive. Careful attention to signs is essential to maintaining the equation's balance and accuracy.

3. Combining Non-Like Terms: A fundamental rule in algebra is that you can only combine like terms. Like terms have the same variable raised to the same power. Attempting to combine terms such as 2x and 6 in the equation 2x + 6 = 7x - 9 would be an error. Ensure you only combine terms that have the same variable and exponent or are constants.

4. Incorrect Order of Operations: The order of operations (PEMDAS/BODMAS) is crucial in simplifying expressions. Failing to adhere to the correct order can lead to errors. Remember to perform operations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). When verifying solutions, strictly following the order of operations is vital.

5. Forgetting to Verify the Solution: As we've emphasized, verifying the solution is a critical step in the problem-solving process. Skipping this step leaves room for undetected errors. Substituting the solution back into the original equation confirms its validity and catches any mistakes made during the solving process. Make it a habit to always verify your solutions.

By being mindful of these common mistakes and consistently practicing careful techniques, you can minimize errors and enhance your proficiency in solving linear equations. Remember, accuracy is just as important as understanding the process.

Conclusion: Mastering Linear Equations

In this comprehensive guide, we've meticulously dissected the process of solving the linear equation 2(x + 3) = 7x - 9. We've explored the fundamental principles of linear equations, walked through a step-by-step solution, emphasized the importance of verifying the solution, and highlighted common mistakes to avoid. By mastering these concepts and techniques, you're well-equipped to tackle a wide array of linear equations with confidence and accuracy.

Linear equations form the bedrock of algebra and are encountered in various mathematical contexts. The ability to solve them efficiently and correctly is crucial for building a strong foundation in mathematics and related fields. The systematic approach we've outlined – distributing, grouping like terms, isolating the variable, and verifying the solution – provides a robust framework for solving linear equations of varying complexity.

Remember, practice is key to mastery. The more you practice solving linear equations, the more comfortable and proficient you'll become. Don't hesitate to revisit this guide and work through additional examples. Embrace challenges and view mistakes as opportunities for learning and growth.

As you continue your mathematical journey, the skills and understanding you've gained in solving linear equations will serve as a valuable asset. They'll enable you to progress to more advanced topics in algebra and beyond. So, keep honing your skills, stay curious, and continue exploring the fascinating world of mathematics. The ability to solve linear equations is not just a mathematical skill; it's a problem-solving tool that can be applied in various aspects of life.