Solving The Equation 3x + 5 = 17 Step-by-Step Guide

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Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more complex mathematical concepts. In this comprehensive guide, we will delve into the step-by-step process of solving the linear equation 3x + 5 = 17. We will break down each step, providing clear explanations and insights to ensure a thorough understanding. This guide is designed for students of all levels, from beginners to those looking to refresh their algebraic skills. We will not only solve this specific equation but also discuss the underlying principles and techniques that can be applied to a wide range of similar problems. By the end of this guide, you will have a solid grasp of how to solve linear equations and be well-equipped to tackle more advanced mathematical challenges. So, let's embark on this mathematical journey together and unlock the secrets of algebraic equation solving!

Understanding the Basics of Algebraic Equations

Before diving into the solution of the equation 3x + 5 = 17, it's essential to grasp the fundamental concepts of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions are composed of variables (usually represented by letters like x, y, or z), constants (numbers), and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an algebraic equation is to find the value(s) of the variable(s) that make the equation true. In other words, we want to determine the value(s) that, when substituted for the variable(s), will make the left-hand side of the equation equal to the right-hand side.

Linear equations, like the one we're addressing, are a specific type of algebraic equation where the highest power of the variable is 1. This means that the variable is not squared, cubed, or raised to any higher power. Linear equations are characterized by their straightforward structure and relatively simple solutions. They can be represented graphically as straight lines, hence the name "linear." Understanding the properties of linear equations is crucial for solving them effectively. We will explore these properties in detail as we progress through the solution process. Remember, the key to mastering algebraic equations lies in understanding the underlying principles and applying them systematically. So, let's move forward and unravel the solution to our equation!

Step-by-Step Solution of 3x + 5 = 17

Now, let's embark on the step-by-step solution of the equation 3x + 5 = 17. Our primary objective is to isolate the variable 'x' on one side of the equation. This means we want to manipulate the equation using algebraic operations until we have 'x' by itself on one side and a numerical value on the other side. To achieve this, we will employ the fundamental principle of equality, which states that we can perform the same operation on both sides of an equation without changing its validity. This principle allows us to add, subtract, multiply, or divide both sides of the equation by the same value, as long as we do it consistently.

Step 1: Isolate the term with 'x'

The first step in solving the equation is to isolate the term containing 'x'. In our case, this term is '3x'. To isolate it, we need to eliminate the constant term '+5' from the left-hand side of the equation. We can do this by subtracting 5 from both sides of the equation. This operation maintains the equality because we are performing the same action on both sides. The equation now becomes:

3x + 5 - 5 = 17 - 5

Simplifying both sides, we get:

3x = 12

Step 2: Solve for 'x'

Now that we have isolated the term '3x', the next step is to solve for 'x' itself. Currently, 'x' is being multiplied by 3. To undo this multiplication and isolate 'x', we need to perform the inverse operation, which is division. We will divide both sides of the equation by 3. Again, this maintains the equality because we are performing the same operation on both sides. The equation becomes:

3x / 3 = 12 / 3

Simplifying both sides, we get:

x = 4

Therefore, the solution to the equation 3x + 5 = 17 is x = 4. This means that when we substitute 4 for 'x' in the original equation, the equation holds true. We have successfully solved for 'x' by systematically isolating it using algebraic operations. In the following sections, we will verify our solution and discuss the implications of this result.

Verifying the Solution

After finding a solution to an equation, it's crucial to verify its correctness. This step ensures that the value we obtained for the variable indeed satisfies the original equation. To verify our solution x = 4 for the equation 3x + 5 = 17, we substitute 4 for 'x' in the original equation and check if the left-hand side equals the right-hand side.

Substituting x = 4 into the equation, we get:

3(4) + 5 = 17

Now, we perform the arithmetic operations on the left-hand side:

12 + 5 = 17

17 = 17

As we can see, the left-hand side of the equation equals the right-hand side when x = 4. This confirms that our solution is correct. Verification is an essential step in problem-solving as it helps prevent errors and ensures the accuracy of the result. By substituting the solution back into the original equation, we can be confident that we have found the correct value for the variable. In the next section, we will discuss the implications of our solution and its significance in the context of linear equations.

Implications and Applications of the Solution

Now that we have successfully solved the equation 3x + 5 = 17 and verified our solution x = 4, let's discuss the implications and applications of this result. The solution x = 4 represents the specific value of 'x' that makes the equation true. In other words, it's the only value that, when substituted for 'x', will make the left-hand side of the equation equal to the right-hand side. This concept is fundamental to understanding the nature of solutions in algebraic equations.

Linear equations, like the one we solved, have a wide range of applications in various fields, including mathematics, science, engineering, and economics. They are used to model relationships between quantities that change at a constant rate. For example, linear equations can be used to represent the cost of a service based on a fixed fee plus a variable cost per unit, the distance traveled by an object moving at a constant speed, or the relationship between supply and demand in economics.

Understanding how to solve linear equations is crucial for solving real-world problems. The techniques we used to solve 3x + 5 = 17, such as isolating the variable and performing inverse operations, can be applied to a wide variety of linear equations. The ability to solve these equations empowers us to make predictions, analyze data, and make informed decisions in various contexts. In the following sections, we will explore more complex equations and problem-solving strategies, building upon the foundation we have established in this guide.

Solving More Complex Equations

Having mastered the solution of the equation 3x + 5 = 17, we can now extend our skills to solving more complex equations. The fundamental principles and techniques we employed remain the same, but the equations may involve additional terms, parentheses, or fractions. Let's consider an example of a slightly more complex equation:

2(x + 3) - 5 = 15

To solve this equation, we need to follow a systematic approach, similar to what we did before, but with some additional steps.

Step 1: Simplify the equation

First, we need to simplify the equation by removing any parentheses or combining like terms. In this case, we have parentheses around the expression '(x + 3)'. To remove the parentheses, we apply the distributive property, which states that a(b + c) = ab + ac. Applying this property to our equation, we get:

2x + 6 - 5 = 15

Now, we can combine the like terms '+6' and '-5' on the left-hand side:

2x + 1 = 15

Step 2: Isolate the term with 'x'

Next, we isolate the term containing 'x', which is '2x'. To do this, we subtract 1 from both sides of the equation:

2x + 1 - 1 = 15 - 1

Simplifying both sides, we get:

2x = 14

Step 3: Solve for 'x'

Finally, we solve for 'x' by dividing both sides of the equation by 2:

2x / 2 = 14 / 2

Simplifying both sides, we get:

x = 7

Therefore, the solution to the equation 2(x + 3) - 5 = 15 is x = 7. This example demonstrates how we can extend our problem-solving skills to more complex equations by applying the same fundamental principles in a systematic manner. In the following sections, we will explore other types of equations and techniques for solving them, further enhancing our mathematical abilities.

Conclusion

In this comprehensive guide, we have explored the process of solving the equation 3x + 5 = 17, along with a broader discussion of algebraic equations and their applications. We began by understanding the basics of algebraic equations and the concept of solving for a variable. We then systematically solved the equation 3x + 5 = 17, breaking down each step and providing clear explanations. We verified our solution to ensure its correctness and discussed the implications and applications of the result.

Furthermore, we extended our skills to solving more complex equations, demonstrating how the same fundamental principles can be applied to a wider range of problems. The key takeaways from this guide are the importance of understanding the underlying principles of algebraic equations, the systematic approach to solving them, and the significance of verification in ensuring accuracy.

By mastering the techniques presented in this guide, you have equipped yourself with valuable tools for solving algebraic equations and tackling mathematical challenges. The journey of mathematical learning is continuous, and we encourage you to explore more advanced topics and problem-solving strategies. With practice and dedication, you can further enhance your mathematical abilities and unlock the power of algebra in various aspects of your life. Remember, mathematics is not just about numbers and equations; it's about logical thinking, problem-solving, and the beauty of abstract concepts. Embrace the challenge and continue your mathematical journey with enthusiasm and curiosity!