Solving 3(t-3) = 5(2t+1) A Step-by-Step Guide

In this article, we will delve into the process of solving the linear equation 3(t-3)=5(2t+1). This equation is a fundamental example of linear equations, which are ubiquitous in mathematics and various scientific and engineering disciplines. Understanding how to solve such equations is crucial for developing a solid foundation in algebra and its applications. We will meticulously walk through each step of the solution, providing clear explanations and justifications along the way. This detailed approach aims to make the solution accessible to readers with varying levels of mathematical background, from students learning basic algebra to anyone looking to refresh their skills. By the end of this article, you will not only know the solution to this specific equation but also gain a deeper understanding of the principles and techniques involved in solving linear equations in general.

The given equation is:

3(t-3)=5(2t+1)

The first step in solving this equation is to eliminate the parentheses. This is achieved by applying the distributive property, which states that a(b+c) = ab + ac. Applying this property to both sides of the equation, we get:

3 * t - 3 * 3 = 5 * 2t + 5 * 1

This simplifies to:

3t - 9 = 10t + 5

Now, we need to isolate the variable 't' on one side of the equation. To do this, we will first move all terms containing 't' to one side and all constant terms to the other side. A common strategy is to move the terms in such a way that the coefficient of 't' remains positive. In this case, we can subtract 3t from both sides of the equation:

3t - 9 - 3t = 10t + 5 - 3t

This simplifies to:

-9 = 7t + 5

Next, we will move the constant term (+5) from the right side to the left side. To do this, we subtract 5 from both sides of the equation:

-9 - 5 = 7t + 5 - 5

This simplifies to:

-14 = 7t

Now, we have the term 7t isolated on the right side. To solve for 't', we need to isolate 't' by dividing both sides of the equation by its coefficient, which is 7:

-14 / 7 = 7t / 7

This simplifies to:

-2 = t

Therefore, the solution to the equation 3(t-3)=5(2t+1) is t = -2.

To ensure the correctness of our solution, it's crucial to verify it by substituting the value of 't' back into the original equation. This process helps us identify any potential errors made during the solution steps. Substituting t = -2 into the original equation 3(t-3)=5(2t+1), we get:

3((-2) - 3) = 5(2(-2) + 1)

Now, we simplify both sides of the equation:

3(-5) = 5(-4 + 1)

-15 = 5(-3)

-15 = -15

Since both sides of the equation are equal, our solution t = -2 is correct. This verification step is a fundamental practice in algebra, reinforcing the accuracy of the solution and demonstrating a thorough understanding of the problem-solving process. It provides confidence in the final answer and solidifies the steps taken to arrive at it.

While the step-by-step solution presented above is a standard method for solving linear equations, there are alternative approaches that can be used. These alternative methods may offer different perspectives or efficiencies depending on the specific equation and the solver's preferences. One such approach involves rearranging the terms in a slightly different order or combining steps. For instance, instead of subtracting 3t from both sides in the initial steps, one could subtract 10t from both sides. While this leads to negative coefficients, the final solution remains the same. Another approach might involve simplifying the equation in multiple smaller steps, which can sometimes reduce the chance of making algebraic errors. Understanding these alternative approaches not only enhances problem-solving skills but also provides a deeper appreciation for the flexibility and adaptability of algebraic techniques. Exploring different methods can also help in identifying the most efficient strategy for solving similar equations in the future, making the solver more proficient and confident in their mathematical abilities.

Solving linear equations is a fundamental skill, but it is also an area where common mistakes can easily occur. Being aware of these pitfalls can significantly improve accuracy and understanding. One frequent mistake is incorrectly applying the distributive property. For example, in the equation 3(t-3)=5(2t+1), a student might mistakenly multiply only the first term inside the parenthesis, resulting in 3t - 3 instead of the correct 3t - 9. Another common error is making sign mistakes when adding or subtracting terms on both sides of the equation. For instance, when moving a term from one side to the other, the sign must be changed. Neglecting this can lead to an incorrect solution. Additionally, errors can occur when dividing or multiplying both sides of the equation by a negative number. It's essential to remember that dividing or multiplying by a negative number will change the sign of all terms. To avoid these mistakes, it's crucial to double-check each step, particularly the application of the distributive property and the manipulation of signs. Verification of the solution by substituting it back into the original equation is also a vital step in catching and correcting errors.

Linear equations are not just abstract mathematical concepts; they have numerous practical applications in real-world scenarios. These applications span various fields, including science, engineering, economics, and everyday life. For example, in physics, linear equations are used to describe motion with constant velocity or acceleration. In engineering, they are used in circuit analysis to determine currents and voltages. In economics, linear equations can model supply and demand relationships or calculate break-even points. Even in everyday situations, linear equations come into play. For instance, calculating the total cost of items with a fixed price per item or determining the time it takes to travel a certain distance at a constant speed involves solving linear equations. The versatility of linear equations makes them an essential tool for problem-solving in diverse contexts. Understanding how to set up and solve linear equations enables individuals to make informed decisions, analyze data, and predict outcomes in various real-life situations. This practical relevance underscores the importance of mastering the techniques discussed in this article.

In this article, we have provided a comprehensive guide to solving the linear equation 3(t-3)=5(2t+1). We meticulously walked through each step, providing clear explanations and justifications. We also emphasized the importance of verification to ensure the accuracy of the solution. Furthermore, we explored alternative approaches, highlighted common mistakes to avoid, and discussed the real-world applications of linear equations. By mastering the techniques discussed here, readers can develop a strong foundation in algebra and confidently tackle similar problems. Linear equations are a fundamental building block in mathematics, and understanding them is essential for success in more advanced topics. We encourage readers to practice solving various linear equations to further solidify their understanding and skills. The ability to solve linear equations accurately and efficiently is a valuable asset in academic pursuits, professional careers, and everyday life.