Solving 5t - 3 = 3t - 5 A Step By Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of algebraic problem-solving. This article delves into the process of solving a specific linear equation: 5t - 3 = 3t - 5. We will break down each step, providing a clear and concise explanation to help you grasp the underlying principles. This detailed guide not only provides the solution but also enhances your understanding of solving similar algebraic problems. Understanding how to solve linear equations is crucial as they appear in various mathematical contexts and real-world applications. This guide will walk you through each step, ensuring you understand not just the answer, but the methodology behind it. Mastering linear equations equips you with a valuable tool for tackling more complex mathematical challenges. Therefore, whether you are a student learning algebra or someone looking to refresh your math skills, this article offers a comprehensive approach to solving the equation 5t - 3 = 3t - 5.

Before diving into the solution, it’s essential 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 when plotted on a graph, they form a straight line. The general form of a linear equation is ax + b = c, where x is the variable, and a, b, and c are constants. Understanding this basic structure helps in identifying and solving linear equations effectively. Linear equations are ubiquitous in mathematics and its applications, serving as the foundation for more advanced concepts. They are used to model relationships between quantities that change at a constant rate. This makes them essential for solving real-world problems in various fields such as physics, economics, and engineering. Recognizing the properties of linear equations, such as the principle that performing the same operation on both sides maintains equality, is crucial for successful problem-solving. This foundation allows you to manipulate and simplify equations to isolate the variable and find its value. In essence, mastering linear equations is a key stepping stone towards more complex algebraic concepts and practical applications.

Our goal is to solve the equation 5t - 3 = 3t - 5 for the variable t. This means we need to isolate t on one side of the equation. To do this, we will use a series of algebraic manipulations. The key principle here is maintaining balance: whatever operation we perform on one side of the equation, we must also perform on the other side. This ensures that the equality remains valid throughout the solution process. The given equation represents a linear relationship, and by applying the properties of equality, we can systematically find the value of t that satisfies the equation. This process involves moving terms across the equals sign, combining like terms, and ultimately isolating the variable. The equation 5t - 3 = 3t - 5 is a classic example of a linear equation, and solving it demonstrates the fundamental techniques used in algebra. The solution not only provides a numerical answer but also reinforces the understanding of algebraic principles. Each step in the solution is carefully chosen to simplify the equation and bring us closer to isolating the variable. By understanding these steps, you can apply similar methods to solve a wide range of linear equations.

Step 1: Grouping Like Terms

The first step in solving 5t - 3 = 3t - 5 is to group like terms. This involves moving all terms containing t to one side of the equation and all constant terms to the other side. To achieve this, we can subtract 3t from both sides of the equation. This will eliminate the 3t term from the right side and bring it to the left side. Subtracting 3t from both sides maintains the equality of the equation, a crucial principle in algebraic manipulation. This step simplifies the equation by consolidating the variable terms on one side, making it easier to isolate t in subsequent steps. The process of grouping like terms is a fundamental technique in solving any algebraic equation. It helps to organize the equation and makes it easier to identify the next steps required for the solution. By carefully moving terms, we ensure that we are progressing towards isolating the variable while maintaining the balance of the equation. This step sets the stage for further simplification and ultimately finding the value of t. Grouping like terms is not just a procedural step; it’s a strategic move that simplifies the equation and makes the solution more accessible.

5t - 3 - 3t = 3t - 5 - 3t

Simplifying this, we get:

2t - 3 = -5

Step 2: Isolating the Variable Term

Now that we have grouped the variable terms on one side, our next step is to isolate the term containing t. In the equation 2t - 3 = -5, the term with t is 2t. To isolate this term, we need to eliminate the constant term on the same side, which is -3. We can do this by adding 3 to both sides of the equation. This operation maintains the equality and moves us closer to isolating t. Adding 3 to both sides effectively cancels out the -3 on the left side, leaving us with just the 2t term. This step is crucial because it simplifies the equation further and brings us closer to finding the value of t. Isolating the variable term is a common technique in solving equations. It allows us to focus on the variable itself without the distraction of other constants. By performing this step, we prepare the equation for the final stage of solving for t. Each step in the solution process builds upon the previous one, and isolating the variable term is a key milestone in this process. This careful manipulation ensures that we are on the right track to finding the solution.

2t - 3 + 3 = -5 + 3

This simplifies to:

2t = -2

Step 3: Solving for t

With the equation now in the form 2t = -2, we are in the final stage of solving for t. To isolate t, we need to undo the multiplication by 2. We can do this by dividing both sides of the equation by 2. Dividing both sides by 2 maintains the equality and isolates t on the left side. This operation is the inverse of multiplication, and it effectively cancels out the 2 that is multiplying t. This final step reveals the value of t that satisfies the original equation. Solving for t is the ultimate goal of the entire process, and this step brings us to that solution. The division operation is straightforward, and it directly leads to the answer. By dividing both sides by 2, we are left with t equal to a specific value. This value is the solution to the linear equation and represents the point where the two expressions in the original equation are equal. This step not only provides the answer but also reinforces the principles of algebraic manipulation and equation solving.

2t / 2 = -2 / 2

This gives us:

t = -1

Therefore, the solution to the equation 5t - 3 = 3t - 5 is t = -1. This means that when t is equal to -1, the left side of the equation is equal to the right side. We can verify this solution by substituting -1 back into the original equation to ensure that it holds true. The solution t = -1 is a specific value that satisfies the equation, and it represents the point of balance between the two expressions. Finding this solution requires careful application of algebraic principles and a systematic approach to equation solving. The value of t = -1 is not just an answer; it is a key piece of information that tells us about the relationship between the variables in the equation. Understanding the solution helps us to interpret the equation and its implications. The solution also demonstrates the effectiveness of the steps we took to solve the equation. By following these steps, we were able to successfully isolate the variable and find its value. The solution t = -1 is the culmination of our efforts and the final answer to the problem.

To ensure the accuracy of our solution, it is always a good practice to verify it. We substitute t = -1 back into the original equation:

5(-1) - 3 = 3(-1) - 5

Simplifying both sides:

-5 - 3 = -3 - 5

-8 = -8

The equation holds true, confirming that t = -1 is indeed the correct solution. Verification is a crucial step in the problem-solving process. It provides a check on our work and ensures that we have not made any errors in our calculations. Substituting the solution back into the original equation allows us to see if the equation remains balanced. If both sides of the equation are equal after substitution, then our solution is correct. This process not only confirms the answer but also reinforces our understanding of the equation and the solution. Verification can also help us to identify any mistakes we may have made during the solution process. If the equation does not hold true after substitution, then we know that we need to go back and review our steps. Therefore, verification is an essential tool for ensuring the accuracy of our solutions.

In conclusion, we have successfully solved the linear equation 5t - 3 = 3t - 5 and found the solution to be t = -1. This process involved grouping like terms, isolating the variable term, and finally solving for t. Each step was carefully explained to provide a clear understanding of the algebraic manipulations involved. We also verified our solution to ensure its accuracy. This step-by-step guide demonstrates the systematic approach to solving linear equations. Mastering these techniques is crucial for tackling more complex algebraic problems. The solution t = -1 is not just a numerical answer; it represents a specific value that satisfies the equation. Understanding how to find this value is a key skill in mathematics. The ability to solve linear equations is fundamental to many areas of math and science. From basic algebra to advanced calculus, linear equations form the foundation for more complex concepts. Therefore, mastering these techniques is an investment in your mathematical skills. This guide has provided a comprehensive approach to solving the equation, and the principles discussed can be applied to a wide range of similar problems. By practicing these techniques, you can build your confidence and improve your problem-solving abilities.