Solving The Equation 4(x+1)=64 And Finding The Value Of X

In this article, we will delve into the mathematical equation 4(x+1)=64 and provide a step-by-step guide on how to solve for the unknown variable, x. This equation falls under the category of basic algebra, a fundamental branch of mathematics that deals with symbols and the rules for manipulating those symbols. Mastering the techniques to solve such equations is crucial for anyone looking to build a strong foundation in mathematics. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide you with a clear and concise explanation. We will break down the equation, discuss the underlying principles, and demonstrate the methods used to arrive at the solution. Furthermore, we will explore the importance of understanding the order of operations and how it applies to solving algebraic equations. By the end of this article, you will not only know the value of x in this specific equation but also gain a deeper understanding of the algebraic concepts involved, enabling you to tackle similar problems with confidence. So, let's embark on this mathematical journey and unravel the mysteries of this equation together. From understanding the distributive property to isolating the variable, each step will be explained in detail, ensuring that you grasp the core concepts and can apply them effectively. Remember, practice is key to mastering any mathematical skill, so feel free to try out similar equations and solidify your understanding.

Understanding the Equation 4(x+1)=64

Before we dive into solving for x, let's first understand the equation 4(x+1)=64. This equation is a linear equation in one variable, which means it involves only one unknown (x) and the highest power of the variable is 1. The equation is composed of several key components: a coefficient (4), a variable (x), a constant (1), and an equality sign (=) connecting two expressions. The expression on the left side, 4(x+1), represents 4 multiplied by the sum of x and 1. The expression on the right side, 64, is a constant value. The goal is to find the value of x that makes the left side of the equation equal to the right side. To do this, we need to isolate x on one side of the equation. This involves performing algebraic operations on both sides of the equation to maintain the equality. Understanding the structure of the equation is the first step towards solving it. It helps us identify the operations that need to be performed and the order in which they should be applied. For instance, recognizing the presence of parentheses indicates that we might need to apply the distributive property. Similarly, identifying the constant terms and coefficients helps us determine which operations will lead to isolating the variable. The beauty of algebra lies in its systematic approach to solving problems. By understanding the underlying principles and applying them methodically, we can unravel even the most complex equations. So, let's move forward and explore the steps involved in solving for x in this equation.

Step-by-Step Solution for x

To solve the equation 4(x+1)=64, we will follow a series of algebraic steps to isolate x. The first step is to apply the distributive property. This property states that a(b+c) = ab + ac. In our equation, we have 4(x+1), so we distribute the 4 to both x and 1. This gives us 4x + 41, which simplifies to 4x + 4. Now our equation looks like this: 4x + 4 = 64. The next step is to isolate the term with x. To do this, we need to get rid of the constant term on the left side, which is +4. We can do this by subtracting 4 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 the equality. So, we subtract 4 from both sides: 4x + 4 - 4 = 64 - 4. This simplifies to 4x = 60. Now we have the term with x isolated on the left side. The final step is to solve for x by dividing both sides of the equation by the coefficient of x, which is 4. So, we divide both sides by 4: 4x / 4 = 60 / 4. This simplifies to x = 15. Therefore, the value of x that satisfies the equation 4(x+1)=64 is 15. We can verify this solution by substituting x = 15 back into the original equation: 4(15+1) = 4(16) = 64, which confirms our solution.

Verification of the Solution

After finding a solution for an equation, it's always a good practice to verify the solution. This ensures that our calculations are correct and that the value we found for the variable indeed satisfies the original equation. To verify our solution for the equation 4(x+1)=64, where we found x = 15, we will substitute 15 in place of x in the original equation. So, we have 4(15+1). First, we perform the operation inside the parentheses: 15 + 1 = 16. Now our expression becomes 4(16). Next, we multiply 4 by 16: 4 * 16 = 64. So, the left side of the equation, 4(x+1), becomes 64 when x = 15. The right side of the equation is already 64. Since both sides of the equation are equal (64 = 64), we have verified that x = 15 is indeed the correct solution. This process of verification is crucial in mathematics as it helps us catch any errors in our calculations and ensures that our solution is accurate. It also reinforces our understanding of the equation and the algebraic principles involved. By verifying our solutions, we gain confidence in our problem-solving abilities and develop a more thorough understanding of the mathematical concepts.

The Value of x is 15

In conclusion, after carefully solving the equation 4(x+1)=64, we have determined that the value of x is 15. This solution was obtained by applying the distributive property, isolating the variable term, and then solving for x by dividing both sides of the equation by the coefficient of x. We also verified our solution by substituting x = 15 back into the original equation and confirming that both sides of the equation were equal. This exercise demonstrates the fundamental principles of algebra and the systematic approach to solving linear equations. The ability to solve such equations is a crucial skill in mathematics and has applications in various fields, including science, engineering, and finance. By understanding the steps involved and practicing regularly, one can master the techniques required to solve algebraic equations with confidence. Remember, the key to success in mathematics is not just memorizing formulas but understanding the underlying concepts and applying them logically. So, continue to practice and explore different types of equations to further enhance your mathematical skills.

Addressing the Second Part of the Question: The Value of x when p is -5

The second part of the question, "The value of x when p is -5 is:" seems to be incomplete or missing context. There is no mention of p in the original equation, 4(x+1)=64, or any other equation provided in the initial problem statement. Therefore, we cannot determine the value of x when p is -5 without additional information or a separate equation that relates x and p. It is possible that there was a typographical error or that part of the question is missing. In such cases, it's important to clarify the question or seek additional information to provide a meaningful answer. In mathematics, it's crucial to have all the necessary information before attempting to solve a problem. Without a clear relationship between x and p, we can only speculate or make assumptions, which would not lead to an accurate solution. If there is another equation or context provided that involves both x and p, we can then use algebraic techniques to solve for x when p is -5. This might involve substitution, elimination, or other methods depending on the nature of the equation. However, based on the information currently available, we cannot proceed with solving for x when p is -5. It is important to always carefully review the problem statement and ensure that all the necessary information is present before attempting to find a solution.