Solving 5(x + 2) = 60 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. It's like deciphering a code, where we aim to uncover the hidden value of a variable, often represented by x. This article will provide a comprehensive guide on how to solve the equation 5(x + 2) = 60. We will break down each step, ensuring clarity and understanding. Mastering this type of equation is crucial for building a solid foundation in algebra and beyond.

The ability to solve equations is not just limited to the classroom; it extends to various real-world scenarios. From calculating expenses to determining optimal quantities in business, equations are ubiquitous. Understanding the mechanics of solving equations empowers you to approach problems logically and find precise solutions. This detailed guide aims to equip you with the knowledge and confidence to tackle similar mathematical challenges.

Solving for x in this equation involves a methodical approach. We will employ the principles of algebra, such as the distributive property and inverse operations, to isolate x on one side of the equation. This process ensures that we maintain the equality throughout, leading us to the correct solution. By the end of this guide, you will be able to solve the equation 5(x + 2) = 60 with ease and apply the same techniques to other algebraic problems. We will cover each step in detail, from the initial distribution to the final verification of the solution. Let’s embark on this mathematical journey together and unlock the value of x!

The first step in solving the equation 5(x + 2) = 60 involves the distributive property. This property states that a(b + c) = ab + ac. In our case, we need to distribute the 5 across the terms inside the parentheses (x + 2). This means we multiply 5 by both x and 2. Applying the distributive property is a fundamental technique in algebra and is essential for simplifying expressions and solving equations.

Multiplying 5 by x gives us 5x, and multiplying 5 by 2 gives us 10. Therefore, when we distribute the 5 across (x + 2), we get 5x + 10. This step transforms the equation from 5(x + 2) = 60 to 5x + 10 = 60. It is crucial to perform this distribution accurately to maintain the equality of the equation and proceed towards the correct solution. The distributive property allows us to remove the parentheses and create a simpler equation that is easier to manipulate.

Understanding and correctly applying the distributive property is crucial not only for solving this specific equation but also for tackling more complex algebraic problems. It forms the basis for simplifying expressions and is a cornerstone of algebraic manipulation. By transforming the equation in this way, we pave the path for isolating x and finding its value. Now that we have distributed the 5, the equation 5x + 10 = 60 is ready for the next step, which involves isolating the term containing x. This is a critical step towards solving for the unknown variable and uncovering its value.

After applying the distributive property, our equation is now 5x + 10 = 60. The next step is to isolate the term containing x, which is 5x. To do this, we need to eliminate the +10 on the left side of the equation. We accomplish this by performing the inverse operation, which is subtraction. The goal is to get 5x by itself on one side of the equation. This process relies on the principle that performing the same operation on both sides of an equation maintains the equality.

To eliminate the +10, we subtract 10 from both sides of the equation. This gives us 5x + 10 - 10 = 60 - 10. Simplifying this, we get 5x = 50. By subtracting 10 from both sides, we have successfully isolated the term with x. This is a critical step in the process of solving for x because it brings us closer to determining the value of the variable. Isolating the variable term is a common technique used in solving various types of equations.

By isolating the term 5x, we have simplified the equation and set the stage for the final step, which is to solve for x. This step demonstrates the importance of inverse operations in algebra. By understanding how to use inverse operations, we can systematically manipulate equations to isolate variables and find solutions. The equation 5x = 50 is now much easier to solve, and we are just one step away from uncovering the value of x. The next step will involve dividing both sides by the coefficient of x.

We have reached the stage where our equation is 5x = 50. To solve for x, we need to isolate x completely. Currently, x is being multiplied by 5. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 5. This step is based on the principle that dividing both sides of an equation by the same non-zero number maintains the equality.

Dividing both sides of the equation 5x = 50 by 5 gives us (5x)/5 = 50/5. Simplifying this, we get x = 10. Therefore, the solution to the equation is x equals 10. This final step is the culmination of the previous steps, where we systematically applied algebraic principles to isolate x and determine its value. Solving for x is the ultimate goal of the equation-solving process.

We have successfully solved the equation 5(x + 2) = 60 and found that x = 10. This process highlights the importance of inverse operations in algebra. By understanding how to use division to undo multiplication, we were able to isolate x and find its value. The solution x = 10 is the number that, when substituted back into the original equation, will make the equation true. This is a crucial concept in algebra, as it confirms the accuracy of our solution. We can now proceed to the final step, which is to verify our solution.

After finding the solution x = 10, it's crucial to verify the solution to ensure accuracy. This step involves substituting the value of x back into the original equation, 5(x + 2) = 60, and checking if both sides of the equation are equal. Verification is an essential step in problem-solving, as it provides confidence in the correctness of the answer. It also helps in identifying any potential errors made during the solving process.

Substituting x = 10 into the original equation, we get 5(10 + 2) = 60. First, we simplify the expression inside the parentheses: 10 + 2 = 12. So, the equation becomes 5(12) = 60. Next, we multiply 5 by 12, which gives us 60. Therefore, we have 60 = 60. Since both sides of the equation are equal, our solution x = 10 is correct.

Verifying the solution is a critical step in solving any equation. It not only confirms the accuracy of the answer but also reinforces the understanding of the equation-solving process. By substituting the value of x back into the original equation and ensuring that both sides are equal, we can be confident in our solution. In this case, the verification step has confirmed that x = 10 is indeed the correct solution to the equation 5(x + 2) = 60. This completes our journey of solving this equation, and we can now confidently state the value of x.

In conclusion, we have successfully solved the equation 5(x + 2) = 60 using a step-by-step approach. We began by applying the distributive property, followed by isolating the term with x, solving for x, and finally, verifying the solution. Each step was crucial in ensuring that we arrived at the correct answer. The entire process showcases the power of algebraic principles in solving equations and revealing the value of unknown variables.

By following these steps, we determined that the value of x in the equation 5(x + 2) = 60 is 10. This solution was confirmed through verification, where we substituted x = 10 back into the original equation and found that both sides were equal. The ability to solve equations like this is fundamental to mathematical proficiency and has wide-ranging applications in various fields.

Mastering equation-solving techniques is not just about finding the answer; it's about developing critical thinking and problem-solving skills. The methodical approach we used in this guide can be applied to a wide variety of algebraic problems. Understanding the underlying principles, such as the distributive property and inverse operations, is key to becoming a confident and capable problem solver. The journey of solving for x in this equation has not only revealed its value but has also reinforced the importance of algebraic principles in mathematical problem-solving.

Therefore, the final answer is x = 10.