Solving The Equation 2x + 5 = 15 A Step-by-Step Guide

Introduction

Hey guys! Let's dive into solving a simple algebraic equation today. We're going to break down how to solve the equation 2x + 5 = 15 step-by-step. Don't worry, it's easier than it looks! Whether you're a student tackling homework or just brushing up on your math skills, this guide will help you understand the process. Understanding basic algebra is crucial as it forms the foundation for more advanced mathematical concepts and has practical applications in various fields, including physics, engineering, and economics. This equation, 2x + 5 = 15, is a linear equation, which means it involves a variable (in this case, x) raised to the power of 1. Solving it involves isolating the variable on one side of the equation to find its value. Linear equations are fundamental in algebra and are used to model a wide range of real-world scenarios. For example, they can represent relationships between quantities like distance and time, cost and quantity, or supply and demand. Mastering the techniques for solving linear equations is therefore an essential skill for anyone studying mathematics or working in a quantitative field. We'll take you through each stage, explaining the logic behind every move, so by the end, you'll be able to solve similar equations with confidence. We will cover the basics of algebraic manipulation, focusing on maintaining the balance of the equation while isolating the variable. Remember, the key to solving equations is to perform the same operation on both sides, ensuring that the equality remains valid. This principle is rooted in the fundamental properties of equality, which state that adding, subtracting, multiplying, or dividing both sides of an equation by the same value does not change the solution. So, let's put on our math hats and get started! Let's start with what the equation means.

Understanding the Equation

The equation 2x + 5 = 15 might look a bit intimidating at first, but let’s break it down. The equation, 2x + 5 = 15, represents a mathematical relationship where the left side, 2x + 5, is equal to the right side, 15. The variable x is an unknown value that we need to determine. The coefficient 2 in front of x means that x is being multiplied by 2. The addition of 5 indicates that 5 is being added to the product of 2 and x. The equals sign (=) is the heart of the equation, showing that the value of the expression on the left side is exactly the same as the value on the right side. In essence, we are looking for a value of x that, when multiplied by 2 and then added to 5, will result in 15. This type of equation is called a linear equation because it involves a variable raised to the power of 1. Linear equations can be visualized as straight lines on a graph, and solving them often involves finding the point where the line intersects a particular value. Understanding the components of the equation is the first step toward solving it. Each term and operation plays a crucial role in determining the value of the unknown variable. By systematically isolating the variable, we can unravel the equation and find the solution. The goal is to manipulate the equation in such a way that we end up with x alone on one side, giving us its value. This involves using inverse operations to undo the operations that are being performed on x. For example, if a number is being added to x, we can subtract that number from both sides of the equation to isolate x. Similarly, if x is being multiplied by a number, we can divide both sides of the equation by that number. So, let’s dive in and start the process of solving this equation step-by-step. We will start by isolating the term with 'x'.

Step 1: Isolate the Term with 'x'

Our main goal is to get x by itself on one side of the equation. To do this, we need to start by isolating the term that contains x, which in this case is 2x. Currently, we have 2x + 5 = 15. To isolate 2x, we need to get rid of the + 5. Remember, in algebra, we use inverse operations to undo operations. The inverse operation of addition is subtraction. So, to get rid of the + 5, we'll subtract 5 from both sides of the equation. Why both sides? Because we need to keep the equation balanced. Think of an equation like a scale: if you add or subtract something from one side, you need to do the same on the other side to keep it level. This is a fundamental principle in algebra – whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain equality. By subtracting 5 from both sides, we are effectively moving the constant term from the left side to the right side, while changing its sign. This is a common technique used in solving equations. It simplifies the equation by grouping like terms together, making it easier to isolate the variable. In this step, we are focusing on isolating the term with x to prepare for the next step, where we will isolate x itself. By systematically removing the constants and coefficients around x, we get closer to finding the value of x that satisfies the equation. This process of isolating the variable is a cornerstone of algebraic problem-solving and is used in a wide variety of mathematical and scientific contexts. So, let's perform this operation and see what the equation looks like after this first step. This sets the stage for further simplification and ultimately finding the solution for x. Now, let's move on to the next step.

So, we subtract 5 from both sides:

2x + 5 - 5 = 15 - 5

This simplifies to:

2x = 10

Step 2: Solve for 'x'

Now we have 2x = 10. We're almost there! The goal is to get x all by itself. Currently, x is being multiplied by 2. To undo this multiplication, we need to use the inverse operation, which is division. So, we'll divide both sides of the equation by 2. Dividing both sides of the equation by the same number is another application of the principle of maintaining equality. Just as with subtraction, whatever we do to one side, we must do to the other to keep the equation balanced. By dividing both sides by 2, we are isolating x on the left side, effectively solving for its value. This step is crucial because it directly leads to the solution of the equation. Once we divide 10 by 2, we will have the value of x that makes the original equation true. This process of dividing by the coefficient of the variable is a standard technique in algebra and is used whenever the variable is being multiplied by a number. It's a straightforward way to undo the multiplication and reveal the value of the variable. In this step, we are completing the process of isolating x and finding the solution to the equation. By dividing both sides by 2, we are effectively undoing the multiplication that was being performed on x, leaving us with x alone on one side of the equation. This final step brings us to the answer, which is the value of x that satisfies the equation 2x + 5 = 15. So, let's carry out this division and find out what x equals.

So, we divide both sides by 2:

(2x) / 2 = 10 / 2

This simplifies to:

x = 5

Conclusion

And there you have it! We've solved the equation 2x + 5 = 15, and we found that x = 5. Isn't that cool? Solving equations like this might seem tricky at first, but with practice, you'll become a pro. Remember, the key is to use inverse operations and keep the equation balanced. Algebraic equations are fundamental tools in mathematics and science, used to model and solve a wide variety of problems. Understanding how to manipulate equations and solve for unknowns is a crucial skill for anyone studying these fields. In this guide, we've walked through the process of solving a linear equation step-by-step, demonstrating how to isolate the variable and find its value. We've emphasized the importance of maintaining equality by performing the same operations on both sides of the equation. We've also shown how to use inverse operations to undo addition, subtraction, multiplication, and division. By mastering these techniques, you'll be well-equipped to tackle more complex equations and mathematical problems. So, keep practicing, keep exploring, and keep solving! The world of mathematics is full of fascinating challenges and discoveries, and every equation you solve is a step forward on your journey. Remember, solving equations is not just about finding the right answer; it's about developing problem-solving skills, logical thinking, and a deeper understanding of the relationships between numbers and variables. So, congratulations on solving your first equation, and keep up the great work!

If you want to check your answer, you can substitute x = 5 back into the original equation:

2 * 5 + 5 = 15

10 + 5 = 15

15 = 15

The equation holds true, so our answer is correct!