Solving 9x - 20 - 4x = 40 Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. Algebraic equations often require us to isolate a variable, such as 'x,' to find its value. This article delves into the step-by-step process of solving the equation 9x - 20 - 4x = 40. We will break down each step, providing clear explanations and insights to help you understand the underlying principles. Whether you are a student learning algebra or someone looking to refresh your math skills, this guide will equip you with the knowledge to confidently tackle similar problems. We will cover simplifying the equation, combining like terms, isolating the variable, and finally, arriving at the solution. So, let's embark on this mathematical journey together and unravel the value of 'x' in the given equation.
Understanding the Basics of Algebraic Equations
Before diving into the specific equation, let's establish a solid foundation by understanding the basics of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions contain variables, which are symbols (usually letters like 'x', 'y', or 'z') that represent unknown values. The primary goal in solving an algebraic equation is to find the value(s) of the variable(s) that make the equation true. This involves manipulating the equation using various operations while maintaining the balance between the two sides.
Key concepts to grasp include terms, coefficients, and constants. A term is a single number or variable, or numbers and variables multiplied together. For example, in the equation 9x - 20 - 4x = 40, 9x, -20, -4x, and 40 are all terms. A coefficient is the numerical factor of a term that contains a variable. In the term 9x, 9 is the coefficient. A constant is a term that has a fixed value and does not contain any variables, such as -20 and 40 in our equation. Understanding these components is crucial for effectively solving equations. We will use these concepts to simplify and solve the equation at hand, ensuring a clear and concise solution process.
Step-by-Step Solution of the Equation 9x - 20 - 4x = 40
Now, let's embark on the step-by-step journey to solve the equation 9x - 20 - 4x = 40. Our goal is to isolate 'x' on one side of the equation to determine its value. We will achieve this by systematically simplifying the equation and applying algebraic principles.
Step 1: Combine Like Terms
The first step in solving the equation is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 9x and -4x are like terms. We can combine them by adding their coefficients: 9x - 4x = 5x. This simplifies the equation to:
5x - 20 = 40
This step is crucial as it reduces the complexity of the equation, making it easier to manage and solve. By combining like terms, we consolidate the variable terms, bringing us closer to isolating 'x'.
Step 2: Isolate the Variable Term
Next, we need to isolate the variable term, which in this case is 5x. To do this, we will add 20 to both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. Adding 20 to both sides gives us:
5x - 20 + 20 = 40 + 20
This simplifies to:
5x = 60
By adding 20 to both sides, we have successfully eliminated the constant term from the left side of the equation, bringing us closer to isolating 'x'.
Step 3: Solve for x
Finally, to solve for 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 5. This will isolate 'x' and give us its value. Dividing both sides by 5, we get:
5x / 5 = 60 / 5
This simplifies to:
x = 12
Therefore, the solution for 'x' in the equation 9x - 20 - 4x = 40 is 12. We have successfully isolated 'x' and determined its value through a series of algebraic manipulations.
Verification of the Solution
After solving for 'x', it is crucial to verify our solution to ensure accuracy. To verify the solution, we substitute the value we found for 'x' back into the original equation. If the equation holds true, then our solution is correct. Let's substitute x = 12 into the original equation:
9x - 20 - 4x = 40
Substituting x = 12, we get:
9(12) - 20 - 4(12) = 40
Now, let's simplify the equation:
108 - 20 - 48 = 40
88 - 48 = 40
40 = 40
Since the equation holds true, our solution x = 12 is indeed correct. This verification step is a vital practice in mathematics to ensure the accuracy of our solutions. By substituting the value back into the original equation, we confirm that our algebraic manipulations were performed correctly and that we have arrived at the correct value for 'x'.
Common Mistakes to Avoid When Solving Algebraic Equations
When solving algebraic equations, it's easy to make mistakes if you're not careful. To ensure accuracy, it's important to be aware of common pitfalls. One frequent error is incorrectly combining like terms. Remember, you can only combine terms that have the same variable raised to the same power. For example, 5x and 3x can be combined, but 5x and 3x² cannot.
Another common mistake is forgetting to apply an operation to both sides of the equation. The golden rule of algebra is that whatever you do to one side of the equation, you must do to the other to maintain equality. For instance, if you add a number to the left side, you must add the same number to the right side.
Incorrectly distributing is another potential error. When an equation involves parentheses, you need to distribute any multiplication or division across all terms inside the parentheses. For example, in the expression 2(x + 3), you need to multiply both x and 3 by 2.
Finally, making arithmetic errors is a common source of mistakes. Simple addition, subtraction, multiplication, or division errors can throw off your entire solution. To avoid this, take your time, double-check your calculations, and consider using a calculator for complex arithmetic.
By being mindful of these common mistakes, you can improve your accuracy and confidence in solving algebraic equations.
Practice Problems to Enhance Your Skills
To truly master the art of solving algebraic equations, practice is essential. Engaging with a variety of problems helps solidify your understanding of the concepts and techniques involved. Let's explore some practice problems that will further enhance your skills.
- Solve for y: 7y + 15 - 3y = 35
- Find the value of z: 12z - 8 = 4z + 24
- Determine a: 5(a - 2) = 3(a + 4)
- Calculate b: 10b - 6 + 2b = 5b + 18
These problems offer a mix of complexities and require you to apply the steps we've discussed, such as combining like terms, isolating the variable, and verifying your solution. Remember to approach each problem systematically, breaking it down into manageable steps. Start by simplifying the equation, then isolate the variable, and finally, solve for the unknown. After finding a solution, always verify your answer by substituting it back into the original equation.
Working through these practice problems will not only improve your equation-solving skills but also boost your confidence in tackling algebraic challenges. Mathematics is a skill that grows with practice, so the more you engage with these problems, the more proficient you will become.
Conclusion
In conclusion, solving algebraic equations is a vital skill in mathematics, and mastering this skill opens doors to more complex mathematical concepts. In this article, we have dissected the process of solving the equation 9x - 20 - 4x = 40 step by step. We began by understanding the basics of algebraic equations, including terms, coefficients, and constants. We then walked through the solution process, which involved combining like terms, isolating the variable term, and finally, solving for x. We found that x = 12 is the solution to the equation.
Moreover, we emphasized the importance of verifying the solution to ensure accuracy. By substituting the value of x back into the original equation, we confirmed that our solution was indeed correct. We also highlighted common mistakes to avoid when solving algebraic equations, such as incorrectly combining like terms, neglecting to apply operations to both sides of the equation, and making arithmetic errors.
To further enhance your skills, we presented a set of practice problems that allow you to apply the techniques discussed in this article. Remember, consistent practice is the key to mastering mathematics. By understanding the underlying principles and diligently practicing, you can confidently tackle a wide range of algebraic equations. So, keep practicing, keep learning, and you'll excel in your mathematical journey.
