Solving 5(x+5) = 85 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill that forms the bedrock of numerous concepts. Equations are the mathematical sentences stating that two expressions are equal, and finding the solution set involves determining the values of the variable(s) that make the equation true. In this guide, we will delve into the process of solving the linear equation 5(x+5) = 85, providing a step-by-step explanation to enhance your understanding and problem-solving abilities. This comprehensive guide aims to equip you with the knowledge and confidence to tackle similar equations effectively. We will break down each step, ensuring clarity and providing insights into the underlying mathematical principles. By the end of this guide, you will not only know how to solve this specific equation but also gain a deeper appreciation for the process of solving equations in general. So, let’s embark on this mathematical journey together and unravel the solution set of the equation 5(x+5) = 85.

Understanding the Equation

Before we dive into the solution, let's first understand the given equation, 5(x+5) = 85. This is a linear equation in one variable, x. The equation states that the expression on the left-hand side, 5(x+5), is equal to the constant 85. Our goal is to find the value(s) of x that satisfy this equation. To do this, we will employ algebraic techniques to isolate x on one side of the equation. Understanding the structure of the equation is crucial for choosing the appropriate solution strategy. The presence of parentheses indicates that we might need to use the distributive property, while the constant term on the right-hand side suggests that we will need to perform inverse operations to isolate x. This initial assessment sets the stage for a systematic approach to solving the equation. By carefully analyzing the equation, we can develop a roadmap for finding the solution set. The key is to break down the equation into smaller, more manageable steps, each of which brings us closer to isolating the variable.

Now, let's proceed with the step-by-step solution of the equation 5(x+5) = 85. We'll break it down into manageable steps, ensuring that each operation is clearly explained.

Step 1: Apply the Distributive Property

Apply the distributive property to eliminate the parentheses. The distributive property states that a(b + c) = ab + ac. In our equation, we have 5(x+5), so we distribute the 5 to both terms inside the parentheses:

5 * x + 5 * 5 = 5x + 25

So, the equation becomes:

5x + 25 = 85

This step is crucial as it simplifies the equation and allows us to proceed with isolating the variable. By distributing the 5, we have transformed the equation into a more manageable form where the variable x is no longer enclosed within parentheses. The distributive property is a fundamental tool in algebra, and its application here is essential for solving the equation. Understanding and correctly applying the distributive property is a key skill for solving various types of equations. This step sets the stage for the subsequent steps, where we will further isolate x and determine its value.

Step 2: Isolate the Term with 'x'

Isolate the term with 'x' by subtracting 25 from both sides of the equation. This maintains the equality while moving the constant term to the right side:

5x + 25 - 25 = 85 - 25

This simplifies to:

5x = 60

This step is essential for isolating the variable term, which is a crucial step in solving for x. By subtracting 25 from both sides, we have effectively removed the constant term from the left side of the equation, leaving only the term containing x. This operation is based on the principle that performing the same operation on both sides of an equation preserves the equality. Understanding this principle is fundamental to solving equations in algebra. The goal here is to get the term with x by itself on one side of the equation, making it easier to isolate x in the next step. This step brings us closer to determining the value of x that satisfies the equation.

Step 3: Solve for 'x'

Solve for 'x' by dividing both sides of the equation by 5. This isolates x and gives us its value:

5x / 5 = 60 / 5

This simplifies to:

x = 12

This final step isolates x and provides the solution to the equation. By dividing both sides by 5, we have effectively removed the coefficient of x, leaving x by itself on the left side of the equation. This operation is the inverse of multiplication, and it is the key to solving for x in this case. The result, x = 12, is the value that satisfies the original equation 5(x+5) = 85. This means that if we substitute 12 for x in the original equation, the equation will hold true. This step completes the process of solving for x, and it is the culmination of the previous steps. Understanding how to isolate a variable by performing inverse operations is a fundamental skill in algebra.

The solution set for the equation 5(x+5) = 85 is {12}. This means that x = 12 is the only value that satisfies the equation. The solution set represents the collection of all values that make the equation true. In this case, there is only one such value, which is 12. To verify this solution, we can substitute 12 for x in the original equation:

5(12 + 5) = 5(17) = 85

Since the equation holds true, we can confidently conclude that the solution set is indeed {12}. Understanding the concept of a solution set is crucial in mathematics, as it represents the complete set of solutions to an equation or system of equations. In this case, the solution set is a singleton set, containing only one element. However, in other equations, the solution set may contain multiple elements or even be an infinite set. The solution set provides a concise way to represent all the solutions to an equation.

In this guide, we have thoroughly explored the process of solving the equation 5(x+5) = 85. We began by understanding the equation, then proceeded with a step-by-step solution, applying the distributive property, isolating the term with 'x', and finally solving for 'x'. We determined that the solution set is {12}. This exercise demonstrates the importance of algebraic techniques in solving equations. By mastering these techniques, you can confidently tackle a wide range of mathematical problems. The ability to solve equations is a fundamental skill in mathematics, and it is essential for various applications in science, engineering, and other fields. We hope this guide has provided you with a clear understanding of the solution process and enhanced your problem-solving abilities. Remember to practice regularly and apply these techniques to different types of equations to further solidify your understanding. The journey of learning mathematics is a continuous process, and every solved equation is a step forward.

Keywords: mathematics, equation, solution set, linear equation, distributive property, isolate x