Solving 2x - 5 = 15 A Step-by-Step Guide

In this comprehensive guide, we will walk through the process of solving the linear equation 2x - 5 = 15. Linear equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for further studies in mathematics and related fields. This guide will break down each step in detail, ensuring a clear understanding of the underlying principles. Whether you're a student tackling homework or someone looking to refresh their algebra skills, this step-by-step approach will help you confidently solve similar equations.

Understanding Linear Equations

Before diving into the solution, it's important to understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called “linear” because they represent a straight line when graphed on a coordinate plane. The general form of a linear equation in one variable is ax + b = c, where x is the variable, and a, b, and c are constants.

In our specific equation, 2x - 5 = 15, we can identify the components: 2 is the coefficient of the variable x, -5 is a constant term, and 15 is the constant on the other side of the equation. The goal of solving a linear equation is to isolate the variable x on one side of the equation. This means we want to manipulate the equation until we have x equal to some value. To achieve this, we use inverse operations, which are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. By applying these operations in the correct order, we can systematically eliminate terms and isolate the variable. Understanding this foundational concept is crucial for solving not just this specific equation, but any linear equation you might encounter. The principles of inverse operations and maintaining equality will be our guiding tools as we proceed through the step-by-step solution. With a solid grasp of these concepts, you'll be well-equipped to tackle more complex algebraic problems in the future. Remember, practice is key to mastery, so work through various examples to build your confidence and proficiency. This equation, 2x - 5 = 15, serves as an excellent starting point for understanding the broader landscape of linear equations and their solutions.

Step 1: Isolate the Term with the Variable

The first step in solving the equation 2x - 5 = 15 is to isolate the term containing the variable x. In this case, the term we want to isolate is 2x. To do this, we need to eliminate the constant term that is on the same side of the equation as the variable term, which is -5. The principle we use here is the addition property of equality, which states that if you add the same number to both sides of an equation, the equation remains balanced. This is a fundamental concept 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. Adding 5 to both sides of the equation will cancel out the -5 on the left side, leaving us with just the term 2x. This process is the inverse operation of subtraction. By adding 5, we are effectively undoing the subtraction. The resulting equation after this step will be simpler and closer to isolating the variable x. Remember, the goal is to gradually peel away the terms and constants surrounding x until x stands alone. This initial step of isolating the variable term is crucial because it sets the stage for the next operation, which will directly involve solving for x. It’s like setting the foundation for a building; a solid first step ensures the stability of the entire process. By carefully applying the addition property of equality, we are making sure that our algebraic manipulations are valid and that the solution we arrive at is accurate. This methodical approach is what distinguishes effective problem-solving in algebra, and it’s a skill that will serve you well in more complex mathematical scenarios.

Adding 5 to both sides of the equation, we get:

2x - 5 + 5 = 15 + 5

Simplifying both sides:

2x = 20

Step 2: Solve for the Variable

Now that we have isolated the term 2x, the next step is to solve for the variable x. The equation we currently have is 2x = 20. This means that 2 multiplied by x equals 20. To find the value of x, we need to undo the multiplication. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by the coefficient of x, which is 2. Just like in the previous step, it's crucial to apply the same operation to both sides of the equation to maintain balance. This is the division property of equality in action, ensuring that we don't alter the fundamental relationship expressed by the equation. Dividing both sides by 2 will isolate x on the left side, giving us the solution. The process of dividing by the coefficient is a standard technique in solving linear equations and is applicable in a wide range of algebraic problems. This step is the culmination of our efforts to isolate and solve for the variable. By performing this division, we are effectively undoing the multiplication that was initially applied to x. The result will be the value of x that satisfies the original equation. This step is not just about finding the answer; it’s about understanding how to manipulate equations using fundamental algebraic principles. It demonstrates the power of inverse operations and the importance of maintaining balance in mathematical manipulations. Solving for x is the ultimate goal in this process, and this step brings us to the final solution, allowing us to understand the specific value that makes the equation true. It’s a moment of clarity where all the preceding steps come together to reveal the answer.

Dividing both sides by 2:

(2x) / 2 = 20 / 2

Simplifying, we get:

x = 10

Step 3: Verify the Solution

After solving for x, it's always a good practice to verify the solution. This step is crucial to ensure that the value we found for x is correct and satisfies the original equation. To verify, we substitute the value of x back into the original equation and check if both sides of the equation are equal. This process acts as a safeguard against any potential errors made during the solving process. Substituting the solution back into the original equation helps confirm that our algebraic manipulations were accurate and that the value we found indeed makes the equation true. Verification is not just a formality; it’s an integral part of problem-solving in mathematics. It helps build confidence in the solution and reinforces the understanding of the equation. This step also provides an opportunity to catch any mistakes that might have been overlooked during the earlier steps. By going through the substitution and simplification process, we can identify inconsistencies and correct them. In essence, verification is a quality control measure in mathematics. It ensures that the answer we've arrived at is not only mathematically sound but also practically applicable. This step is especially important when dealing with more complex equations where the chances of making a mistake are higher. The act of verifying also deepens the understanding of the relationship between the variable and the equation. It solidifies the concept that the solution is a value that makes the equation a true statement. By consistently verifying solutions, one develops a more thorough and reliable approach to problem-solving in algebra and beyond. Therefore, this final step of verification is a valuable habit to cultivate in the journey of mathematical learning.

Substituting x = 10 into the original equation 2x - 5 = 15:

2(10) - 5 = 15

Simplifying:

20 - 5 = 15

15 = 15

Since both sides of the equation are equal, the solution x = 10 is correct.

Conclusion

In this step-by-step guide, we have successfully solved the equation 2x - 5 = 15. By following the principles of isolating the variable term, applying inverse operations, and verifying the solution, we have demonstrated a clear and effective method for solving linear equations. This approach not only provides the correct answer but also reinforces the fundamental concepts of algebra. Remember, practice is key to mastering these skills, so continue to work through various examples to build your confidence and proficiency in solving linear equations. The ability to solve linear equations is a cornerstone of mathematical understanding and is essential for tackling more advanced topics in algebra and beyond. The methodical approach outlined here, from isolating the variable term to verifying the solution, is a valuable framework for problem-solving in mathematics in general. By consistently applying these steps, you can approach any linear equation with clarity and confidence. The understanding gained from solving equations like 2x - 5 = 15 extends to a wide range of mathematical and real-world applications. The principles of balancing equations and using inverse operations are not just abstract concepts; they are tools that can be used to model and solve practical problems in various fields. Therefore, mastering these skills is an investment in your overall mathematical literacy and your ability to apply mathematical thinking to diverse situations. The journey of learning mathematics is often about building upon foundational skills, and solving linear equations is undoubtedly one of those crucial foundations. Keep practicing, keep exploring, and keep building your mathematical toolkit.

By mastering this process, you'll be well-equipped to tackle more complex algebraic problems.