Solving For X In The Equation 2(x+4)-1=19

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving for x in the equation 2(x+4)-1=19. We will break down each step with detailed explanations, ensuring a clear understanding of the algebraic principles involved. Whether you're a student tackling algebra problems or simply seeking to refresh your math skills, this guide will provide a comprehensive approach to solving this type of equation.

Understanding the Equation

Before we dive into the solution, let's first understand the equation we're dealing with: 2(x+4)-1=19. This equation involves several arithmetic operations and algebraic concepts, including multiplication, addition, subtraction, and the distributive property. Our goal is to isolate x on one side of the equation to determine its value. This involves reversing the order of operations and applying inverse operations to both sides of the equation to maintain balance. Understanding the structure of the equation and the order of operations is crucial for solving it accurately.

Step-by-Step Solution

Let's embark on the step-by-step solution to find the value of x. We will meticulously dissect each operation and provide clear reasoning behind every step.

1. Apply the Distributive Property

The first step in solving this equation is to apply the distributive property to the term 2(x+4). The distributive property states that a(b+c) = ab + ac. Applying this property, we multiply 2 by both x and 4:

2(x+4) = 2 * x + 2 * 4 = 2x + 8

Now, the equation becomes:

2x + 8 - 1 = 19

The distributive property is a fundamental concept in algebra, allowing us to simplify expressions and equations by removing parentheses. It's essential to remember to multiply the term outside the parentheses by every term inside the parentheses.

2. Combine Like Terms

Next, we simplify the left side of the equation by combining like terms. In this case, we have the constants 8 and -1. Combining these terms gives us:

8 - 1 = 7

So, the equation now looks like this:

2x + 7 = 19

Combining like terms is a crucial step in simplifying equations, as it reduces the number of terms and makes the equation easier to solve. Identify the constants, variables with the same exponent, and combine them to streamline the equation.

3. Isolate the Variable Term

Our next goal is to isolate the term with x (2x) on one side of the equation. To do this, we need to eliminate the constant term (+7) from the left side. We can achieve this by subtracting 7 from both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain equality:

2x + 7 - 7 = 19 - 7

This simplifies to:

2x = 12

Subtracting the same value from both sides of the equation is a fundamental algebraic principle. It ensures that the equation remains balanced while allowing us to isolate the variable term.

4. Solve for x

Now that we have 2x = 12, we need to solve for x. This means we need to isolate x by dividing both sides of the equation by the coefficient of x, which is 2:

(2x) / 2 = 12 / 2

This gives us:

x = 6

Dividing both sides of the equation by the coefficient of the variable is the final step in isolating x and finding its value. This operation effectively undoes the multiplication, leaving us with the solution for x.

The Solution

Therefore, the solution to the equation 2(x+4)-1=19 is x = 6. This means that when we substitute 6 for x in the original equation, the equation holds true.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the value of x back into the original equation. Let's substitute x = 6 into the equation 2(x+4)-1=19:

2(6+4) - 1 = 19

First, we solve the expression inside the parentheses:

6 + 4 = 10

Now, substitute this back into the equation:

2(10) - 1 = 19

Next, perform the multiplication:

2 * 10 = 20

So, the equation becomes:

20 - 1 = 19

Finally, perform the subtraction:

19 = 19

Since the left side of the equation equals the right side, our solution x = 6 is correct. Verifying the solution is a crucial step in problem-solving, as it confirms the accuracy of our calculations and reasoning.

Alternative Approaches

While we've provided a step-by-step solution, there might be alternative approaches to solving the same equation. For instance, one could choose to add 1 to both sides of the equation first before applying the distributive property. However, the order of operations generally dictates that we address the distributive property before addition or subtraction. Nevertheless, exploring alternative methods can enhance your problem-solving skills and provide a deeper understanding of algebraic principles. Alternative approaches can sometimes offer a more efficient or intuitive way to solve an equation, depending on the individual's preference and understanding of mathematical concepts.

Common Mistakes to Avoid

When solving equations, it's crucial to avoid common mistakes that can lead to incorrect solutions. One frequent error is forgetting to distribute the term outside the parentheses to all terms inside the parentheses. Another mistake is failing to perform the same operation on both sides of the equation, disrupting the balance and leading to an incorrect result. Additionally, errors in arithmetic, such as incorrect addition or subtraction, can also lead to wrong answers. To avoid these mistakes, it's essential to pay close attention to detail, double-check your work, and understand the fundamental principles of algebra. Being mindful of these common pitfalls can significantly improve your accuracy in solving equations.

Conclusion

In conclusion, solving for x in the equation 2(x+4)-1=19 involves applying fundamental algebraic principles such as the distributive property, combining like terms, and performing inverse operations to isolate the variable. By following the step-by-step solution outlined in this guide, you can confidently solve similar equations. Remember to verify your solution to ensure accuracy and avoid common mistakes. Mastering these techniques will strengthen your algebraic skills and enable you to tackle more complex mathematical problems. The ability to solve equations is a cornerstone of mathematics, and with practice and understanding, you can develop proficiency in this essential skill.