Solving 4x - 6 + 2x = 18 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. The goal of solving an equation is to find the value(s) of the unknown variable(s) that make the equation true. Among the various types of equations, linear equations hold a significant position due to their simplicity and widespread applications. Linear equations are equations in which the highest power of the variable is 1. They can be expressed in the general form ax + b = c, where a, b, and c are constants, and x is the variable.

To effectively solve linear equations, a systematic approach is essential. This approach typically involves the following steps:

1. Simplification: Begin by simplifying both sides of the equation. This may involve combining like terms, distributing coefficients, and eliminating parentheses.
2. Isolating the Variable: The next step is to isolate the variable term on one side of the equation. This is achieved by performing the same operation on both sides of the equation to maintain equality. Common operations include adding or subtracting constants, and multiplying or dividing by coefficients.
3. Solving for the Variable: Once the variable term is isolated, solve for the variable by performing the inverse operation. For example, if the variable is multiplied by a constant, divide both sides of the equation by that constant.
4. Verification: After obtaining a solution, it is crucial to verify its correctness. This is done by substituting the solution back into the original equation and checking if it satisfies the equation. If the equation holds true, the solution is correct.

Understanding these steps provides a solid foundation for tackling linear equations. Let's delve into the specific equation at hand and apply these principles to arrive at the solution.

Deciphering the Equation: 4x - 6 + 2x = 18

In this section, we will dissect the equation 4x - 6 + 2x = 18, applying the principles of solving linear equations to determine the value of x that satisfies the equation. This process involves simplifying the equation, isolating the variable, and ultimately solving for x. Each step is crucial in ensuring we arrive at the correct solution. The ability to solve such equations is a cornerstone of mathematical literacy, with applications spanning various fields from science and engineering to economics and finance.

Our initial equation is 4x - 6 + 2x = 18. The first step in solving this equation is to simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power. In this equation, 4x and 2x are like terms, as they both contain the variable x raised to the power of 1. Combining these terms, we add their coefficients: 4x + 2x = 6x. This simplifies our equation to 6x - 6 = 18. This simplification makes the equation easier to work with and brings us closer to isolating the variable x.

Now that we've simplified the equation, the next step is to isolate the variable term. The variable term in this equation is 6x. To isolate it, we need to get rid of the constant term, which is -6, on the left side of the equation. We can do this by adding 6 to both sides of the equation. Adding the same value to both sides maintains the equality of the equation, a fundamental principle in algebra. So, we add 6 to both sides: 6x - 6 + 6 = 18 + 6. This simplifies to 6x = 24. We have now successfully isolated the variable term on one side of the equation.

With the variable term isolated, the final step is to solve for the variable x. The equation 6x = 24 tells us that 6 times x equals 24. To find the value of x, we need to undo the multiplication by 6. We can do this by dividing both sides of the equation by 6. Again, dividing both sides by the same non-zero value maintains the equality. So, we divide both sides by 6: (6x) / 6 = 24 / 6. This simplifies to x = 4. Therefore, the solution to the equation 4x - 6 + 2x = 18 is x = 4. This means that when x is equal to 4, the equation holds true.

By following these steps – simplifying, isolating the variable, and solving for the variable – we have successfully solved the equation. This process demonstrates the power of algebraic manipulation in finding solutions to mathematical problems. Understanding these steps is not only crucial for solving linear equations but also forms a foundation for tackling more complex equations in higher mathematics.

Verifying the Solution: Ensuring Accuracy

After obtaining a solution to an equation, it is essential to verify its correctness. This process involves substituting the obtained value back into the original equation and checking if it satisfies the equation. Verification is a crucial step in problem-solving as it helps to identify potential errors made during the solution process. It ensures that the solution obtained is indeed a valid solution to the equation. In the context of mathematical problem-solving, accuracy is paramount, and verification serves as a safeguard against inaccuracies.

To verify the solution x = 4 for the equation 4x - 6 + 2x = 18, we substitute x = 4 back into the original equation. This means replacing every instance of x in the equation with the value 4. The original equation is 4x - 6 + 2x = 18. Substituting x = 4, we get: 4(4) - 6 + 2(4) = 18. This substitution allows us to evaluate the left-hand side of the equation and check if it equals the right-hand side.

Now, we simplify the left-hand side of the equation following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). First, we perform the multiplications: 4(4) = 16 and 2(4) = 8. So, the equation becomes: 16 - 6 + 8 = 18. Next, we perform the addition and subtraction from left to right: 16 - 6 = 10, and then 10 + 8 = 18. This gives us: 18 = 18. The left-hand side of the equation simplifies to 18, which is equal to the right-hand side of the equation.

Since substituting x = 4 into the original equation results in a true statement (18 = 18), we can conclude that our solution is correct. This verification step confirms that x = 4 is indeed the value that satisfies the equation 4x - 6 + 2x = 18. The verification process not only validates our solution but also reinforces our understanding of the equation and the steps involved in solving it. It is a crucial step in ensuring the accuracy and reliability of our mathematical work.

Implications and Applications of Solving Linear Equations

Solving linear equations is not merely an academic exercise; it has profound implications and widespread applications in various fields. The ability to solve linear equations is a fundamental skill that underpins many aspects of science, engineering, economics, and everyday life. Linear equations provide a framework for modeling and solving real-world problems, making them an indispensable tool in quantitative analysis. Understanding the principles behind solving linear equations opens doors to a deeper understanding of mathematical modeling and problem-solving.

In science and engineering, linear equations are used to model a wide range of phenomena, from the motion of objects to the flow of electricity in circuits. For example, Ohm's Law, a fundamental principle in electrical engineering, is a linear equation that relates voltage, current, and resistance. Similarly, in physics, linear equations are used to describe the motion of objects under constant acceleration. Solving these equations allows scientists and engineers to predict the behavior of systems and design solutions to complex problems. The accuracy of these models often depends on the precise solutions of linear equations.

Economics and finance also heavily rely on linear equations. Supply and demand models, which are used to analyze market equilibrium, often involve solving systems of linear equations. Financial analysts use linear equations to model investment returns and calculate loan payments. Understanding linear relationships is crucial for making informed decisions in financial planning and economic forecasting. The ability to solve these equations accurately is essential for making sound financial judgments and predictions.

In everyday life, linear equations are used in various practical situations. Calculating the cost of items at a store, determining the amount of ingredients needed for a recipe, or figuring out travel time are all examples of problems that can be solved using linear equations. For instance, if you know the hourly rate and the number of hours worked, you can use a linear equation to calculate your earnings. These applications highlight the importance of mathematical literacy in navigating daily life. The ability to apply linear equations in everyday situations empowers individuals to make informed decisions and solve practical problems.

Moreover, the skills developed in solving linear equations are transferable to more complex mathematical concepts. The logical reasoning and problem-solving techniques used in solving linear equations form a foundation for understanding more advanced topics such as calculus, differential equations, and linear algebra. Mastering linear equations is a crucial stepping stone for anyone pursuing further studies in mathematics or related fields. The ability to manipulate equations and isolate variables is a fundamental skill that extends beyond linear equations and into more advanced mathematical disciplines.

In conclusion, solving linear equations is a fundamental skill with far-reaching implications and applications. From scientific modeling to financial analysis and everyday problem-solving, linear equations provide a powerful tool for understanding and interacting with the world around us. The ability to solve these equations accurately and efficiently is essential for success in various academic and professional fields. The principles and techniques learned in solving linear equations form a solid foundation for further mathematical studies and problem-solving endeavors.

In summary, the equation 4x - 6 + 2x = 18 has a solution of x = 4, which we determined through a step-by-step process of simplification, isolation, and solving for the variable. We then verified this solution by substituting it back into the original equation, confirming its accuracy. This process highlights the importance of a systematic approach to solving linear equations, a skill that is crucial not only in mathematics but also in various other fields.

The ability to solve linear equations is a fundamental skill that underpins many aspects of science, engineering, economics, and everyday life. Linear equations provide a framework for modeling and solving real-world problems, making them an indispensable tool in quantitative analysis. Understanding the principles behind solving linear equations opens doors to a deeper understanding of mathematical modeling and problem-solving. The applications of linear equations are vast and varied, ranging from calculating expenses to designing complex systems. Mastering these skills empowers individuals to tackle a wide range of challenges and make informed decisions.

Moreover, the skills developed in solving linear equations are transferable to more complex mathematical concepts. The logical reasoning and problem-solving techniques used in solving linear equations form a foundation for understanding more advanced topics. Mastering linear equations is a crucial stepping stone for anyone pursuing further studies in mathematics or related fields. The ability to manipulate equations and isolate variables is a fundamental skill that extends beyond linear equations and into more advanced mathematical disciplines.

Therefore, the significance of mastering linear equations cannot be overstated. It is a skill that is not only valuable in academic settings but also essential for success in various professional fields and everyday situations. By understanding the principles and techniques involved in solving linear equations, individuals can develop their problem-solving abilities and gain a deeper appreciation for the power and beauty of mathematics. The confidence gained from successfully solving equations can inspire further exploration of mathematical concepts and applications. This mastery serves as a foundation for lifelong learning and problem-solving.

In conclusion, the solution to the equation 4x - 6 + 2x = 18 is x = 4, and the process of finding this solution underscores the importance of mastering linear equations for a wide range of applications and future mathematical endeavors.