Mastering Linear Equation: 2(11t + 1.5t) = 12 - 5t

Welcome to our deep dive into solving linear equations, a fundamental skill in mathematics that opens doors to understanding more complex concepts. Today, we're going to tackle a specific example: 2(11t + 1.5t) = 12 - 5t. This equation might look a little intimidating at first glance with its parentheses and multiple terms, but by breaking it down step-by-step, we'll reveal the straightforward process of finding the value of 't'. Linear equations are the bedrock of algebra, appearing everywhere from simple arithmetic problems to sophisticated scientific models. Understanding how to manipulate and solve them is not just about getting the right answer; it's about developing logical thinking and problem-solving skills that are transferable to countless other areas of your life. We'll guide you through each operation, explaining the 'why' behind each move, ensuring you not only solve this particular equation but also gain a solid grasp of the principles involved. So, grab a pen and paper, and let's embark on this mathematical journey together. We'll cover simplifying expressions, isolating the variable, and checking our final answer, making sure you feel confident and capable by the end of this guide. This process is crucial for anyone studying algebra, as it forms the basis for understanding functions, graphing, and solving systems of equations.

Step 1: Simplify the Equation

Our journey begins with simplifying the given equation: 2(11t + 1.5t) = 12 - 5t. The first thing we notice is the expression inside the parentheses on the left side: 11t + 1.5t. These are like terms because they both contain the variable 't' raised to the same power (which is 1, implicitly). Combining like terms is a fundamental algebraic manipulation. When we add 11t and 1.5t, we are essentially adding their coefficients: 11 + 1.5 = 12.5. So, the expression inside the parentheses simplifies to 12.5t. Now, our equation looks like this: 2(12.5t) = 12 - 5t. The next step on the left side is to distribute the 2 to the 12.5t. This means we multiply 2 by 12.5t, which gives us 2 * 12.5 * t = 25t. So, the simplified left side of our equation is 25t. The right side, 12 - 5t, remains as it is for now, as there are no like terms to combine or operations to perform. After this initial simplification, our equation has transformed into a much more manageable form: 25t = 12 - 5t. This step is critical because it reduces the complexity of the equation, making it easier to isolate the variable 't' in the subsequent steps. Remember, the goal of simplifying is to get the equation into its most basic form without altering its equality. We achieve this by applying the order of operations (PEMDAS/BODMAS) and combining like terms. Consistently applying these rules will ensure accuracy in solving any linear equation. This initial simplification is where many errors can occur if not done carefully, so taking your time here is paramount.

Step 2: Isolate the Variable 't'

Now that we have the simplified equation 25t = 12 - 5t, our next major objective is to get all the terms containing the variable 't' onto one side of the equation and all the constant terms onto the other side. This process is known as isolating the variable. To do this, we need to move the -5t term from the right side of the equation to the left side. The opposite operation of subtracting 5t is adding 5t. Therefore, we will add 5t to both sides of the equation to maintain the balance: 25t + 5t = 12 - 5t + 5t. On the left side, 25t + 5t combines to 30t. On the right side, -5t + 5t cancels each other out, resulting in 0, leaving just the constant term 12. Our equation now becomes 30t = 12. We are one step closer to finding the value of 't'! The principle here is that whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side to keep the equation true. Think of an equality sign as a balancing scale; if you add weight to one side, you must add the same weight to the other to keep it level. We've successfully gathered all the 't' terms on the left. The next step, which we'll cover in the following section, will be to completely isolate 't' by removing the coefficient 30.

Step 3: Solve for 't'

We've reached the penultimate step in solving our linear equation. We currently have 30t = 12. To find the value of a single 't', we need to undo the multiplication of 30 by t. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by 30: 30t / 30 = 12 / 30. On the left side, 30t / 30 simplifies to just t, as anything divided by itself is 1. On the right side, we have 12 / 30. This fraction can be simplified. Both 12 and 30 are divisible by their greatest common divisor, which is 6. So, 12 ÷ 6 = 2 and 30 ÷ 6 = 5. Therefore, 12 / 30 simplifies to 2/5. Our solution is t = 2/5. This means that when 't' is equal to 2/5, the original equation will hold true. It's important to express your answer in its simplest form, whether as a fraction or a decimal. In this case, 2/5 can also be written as 0.4. Presenting the answer clearly is key.

Step 4: Verify Your Solution

To ensure our calculations are correct, we must verify our solution by substituting the value we found for 't' back into the original equation: 2(11t + 1.5t) = 12 - 5t. We found that t = 2/5 (or 0.4). Let's substitute this value into the equation.

Left side: 2(11 * (2/5) + 1.5 * (2/5)) 2(22/5 + 3/5) (since 1.5 = 3/2, and (3/2)*(2/5) = 6/10 = 3/5) 2(25/5) 2(5) 10

Right side: 12 - 5 * (2/5) 12 - 10/5 12 - 2 10

Since the left side (10) equals the right side (10), our solution t = 2/5 is correct! This verification step is crucial in mathematics. It builds confidence in your answer and helps catch any arithmetic errors you might have made along the way. Always take the time to plug your solution back into the original equation. It's a small extra step that guarantees accuracy and reinforces your understanding of the problem. This meticulous approach is what separates good problem-solvers from great ones.

Conclusion

Solving linear equations like 2(11t + 1.5t) = 12 - 5t is a fundamental skill that empowers you to tackle more complex mathematical challenges. We've walked through the process step-by-step: simplifying the equation by combining like terms and distributing, isolating the variable 't' by using inverse operations, solving for 't' by division, and finally, verifying our solution by substituting it back into the original equation. Each step relies on consistent application of algebraic principles, ensuring accuracy and a thorough understanding. Remember, practice is key to mastering any mathematical concept. The more equations you solve, the more intuitive these steps will become. Linear equations are not just abstract mathematical constructs; they are powerful tools used in countless real-world applications, from calculating trajectories in physics to modeling financial growth. By mastering these foundational skills, you equip yourself with the ability to understand and analyze the world around you in a more quantitative way. Keep practicing, and don't hesitate to revisit these steps whenever you encounter a new equation. For further exploration into the fascinating world of algebra and linear equations, you can visit Khan Academy's extensive resources on algebra or delve into the principles of algebra at the Art of Problem Solving website.