Solving 5x + 9 - 6x - 13 + 8x = -18 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the step-by-step process of solving the linear equation 5x + 9 - 6x - 13 + 8x = -18. We will break down each step, providing clear explanations and insights to ensure a comprehensive understanding. Whether you are a student grappling with algebra or simply seeking to refresh your mathematical prowess, this guide will equip you with the knowledge and confidence to tackle similar problems.
1. Understanding Linear Equations
Before we embark on solving the equation, it is crucial to grasp the concept of linear equations. 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 is ax + b = 0, where 'a' and 'b' are constants, and 'x' is the variable. In our case, the equation 5x + 9 - 6x - 13 + 8x = -18 fits this description, making it a linear equation. Understanding this foundation is crucial as it dictates the methods we employ to find the solution. Key characteristics of linear equations include the presence of a single variable raised to the power of one, and the absence of any exponents or other complex functions acting on the variable. This simplicity allows us to use straightforward algebraic manipulations to isolate the variable and determine its value.
2. Simplifying the Equation
The initial step in solving any equation is simplification. This involves combining like terms to reduce the equation to its simplest form. In our equation, 5x + 9 - 6x - 13 + 8x = -18, the like terms are the 'x' terms (5x, -6x, and 8x) and the constant terms (9 and -13). Let's start by combining the 'x' terms: 5x - 6x + 8x. This simplifies to (5 - 6 + 8)x, which equals 7x. Next, we combine the constant terms: 9 - 13. This simplifies to -4. Now, we can rewrite the equation in its simplified form: 7x - 4 = -18. This simplified form is much easier to work with and sets the stage for isolating the variable 'x'. The process of simplification is a cornerstone of equation-solving, ensuring that we are working with the most manageable form of the equation. By accurately combining like terms, we reduce the complexity and pave the way for a straightforward solution.
3. Isolating the Variable Term
Isolating the variable term is the next critical step in solving the equation. Our goal is to get the term containing 'x' (which is 7x in our simplified equation) by itself on one side of the equation. To do this, we need to eliminate any other terms present on the same side. In our equation, 7x - 4 = -18, we have a constant term, -4, on the left side along with the 7x term. To eliminate -4, we perform the inverse operation, which is adding 4 to both sides of the equation. This maintains the balance of the equation, a fundamental principle in algebra. Adding 4 to both sides, we get: 7x - 4 + 4 = -18 + 4. This simplifies to 7x = -14. Now, we have successfully isolated the variable term, 7x, on the left side of the equation. This isolation is a pivotal moment in the solving process, bringing us closer to determining the value of 'x'. By understanding and applying the principle of inverse operations, we can effectively isolate the variable term in any linear equation.
4. Solving for x
With the variable term isolated, the final step is to solve for 'x'. In our equation, we have 7x = -14. This means that 7 times 'x' equals -14. To find the value of 'x', we need to undo the multiplication by 7. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 7. This maintains the equation's balance, ensuring that the solution remains accurate. Dividing both sides by 7, we get: (7x) / 7 = -14 / 7. This simplifies to x = -2. Therefore, the solution to the equation 5x + 9 - 6x - 13 + 8x = -18 is x = -2. This final step demonstrates the power of inverse operations in isolating and solving for the unknown variable. By dividing both sides of the equation by the coefficient of 'x', we successfully determined the value of 'x', completing the solution process.
5. Verifying the Solution
In mathematics, verification is a crucial step to ensure the accuracy of the solution. To verify our solution, x = -2, we substitute this value back into the original equation: 5x + 9 - 6x - 13 + 8x = -18. Substituting x = -2, we get: 5(-2) + 9 - 6(-2) - 13 + 8(-2) = -18. Now, we simplify each term: -10 + 9 + 12 - 13 - 16 = -18. Combining the terms, we get: -1 + 12 - 13 - 16 = -18, which further simplifies to: 11 - 13 - 16 = -18, and then to: -2 - 16 = -18. Finally, we have: -18 = -18. Since the left side of the equation equals the right side, our solution, x = -2, is verified. This process of substitution and simplification confirms that our solution is correct. Verification is an essential practice in mathematics, providing confidence in the accuracy of the result and highlighting the importance of careful calculations.
6. Common Mistakes to Avoid
When solving linear equations, it's easy to fall into common traps. One frequent error is incorrectly combining like terms. For instance, mixing 'x' terms with constant terms. Remember, only terms with the same variable and exponent can be combined. Another common mistake is not applying the same operation to both sides of the equation. Whether it's addition, subtraction, multiplication, or division, maintaining balance is key. Forgetting to distribute a negative sign is also a pitfall. When a negative sign precedes a parenthesis, ensure you apply it to every term inside. Lastly, errors in basic arithmetic can derail the entire process. Double-check your calculations, especially when dealing with negative numbers. By being mindful of these common mistakes, you can improve your accuracy and confidence in solving linear equations. Consistent practice and attention to detail are your best defenses against these errors.
7. 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, from science and engineering to economics and finance. For example, in physics, linear equations can describe the relationship between distance, speed, and time. In economics, they can model supply and demand curves. In finance, they are used to calculate simple interest and loan payments. Linear equations are also fundamental in computer graphics, where they are used to draw lines and shapes on the screen. Understanding and solving linear equations is therefore a valuable skill that extends far beyond the classroom. Their versatility makes them an indispensable tool for problem-solving in many aspects of life. By mastering linear equations, you equip yourself with a powerful tool for analyzing and understanding the world around you.
8. Tips and Tricks for Solving Linear Equations
To enhance your problem-solving skills, here are some tips and tricks for solving linear equations. First, always simplify the equation as much as possible before attempting to isolate the variable. This reduces the complexity and the chance of errors. Second, remember the order of operations (PEMDAS/BODMAS) when simplifying expressions. Third, use the properties of equality to maintain balance; whatever you do to one side, do to the other. Fourth, if you encounter fractions, multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate them. Fifth, always check your solution by substituting it back into the original equation. This ensures accuracy and helps identify any mistakes. Sixth, practice regularly to build your confidence and speed. Solving linear equations is a skill that improves with practice. By incorporating these tips and tricks into your problem-solving routine, you can become more efficient and accurate in solving linear equations.
9. Conclusion
In conclusion, solving the linear equation 5x + 9 - 6x - 13 + 8x = -18 involves a series of logical steps: simplifying the equation, isolating the variable term, solving for x, and verifying the solution. By understanding the underlying principles and avoiding common mistakes, you can confidently tackle similar problems. Linear equations are a fundamental concept in mathematics with wide-ranging applications in various fields. Mastering this skill not only enhances your mathematical abilities but also equips you with a valuable tool for problem-solving in real-world scenarios. Remember, practice is key to success. The more you practice, the more comfortable and proficient you will become in solving linear equations. So, keep practicing, and you will undoubtedly excel in this essential mathematical skill.
