Solving For T In The Equation 3t - 2(t + 12) = 11

Introduction: Unveiling the Solution to a Mathematical Puzzle

In the realm of mathematics, linear equations serve as fundamental building blocks for more complex concepts. These equations, characterized by a variable raised to the power of one, provide a framework for modeling and solving real-world problems. Today, we embark on a journey to unravel the solution to a specific linear equation: 3t - 2(t + 12) = 11. This equation presents a classic example of how algebraic manipulation can lead us to the value of an unknown variable, 't'. By employing a step-by-step approach, we will systematically isolate 't' and uncover its numerical value. This process not only demonstrates the power of algebraic techniques but also reinforces our understanding of fundamental mathematical principles.

The core concept at play here is the principle of equality. This principle dictates that we can perform the same mathematical operation on both sides of an equation without altering its fundamental balance. Whether it's adding, subtracting, multiplying, or dividing, maintaining equality is paramount in solving equations. We'll also leverage the distributive property, a cornerstone of algebra that allows us to expand expressions involving parentheses. The distributive property states that a(b + c) = ab + ac, a rule that will prove invaluable in simplifying our equation. By mastering these concepts and techniques, we equip ourselves with the tools to tackle a wide range of mathematical challenges.

As we delve into the solution, we'll encounter the importance of order of operations. This set of rules, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures that mathematical expressions are evaluated consistently. Adhering to the order of operations is crucial for arriving at the correct answer. Furthermore, we'll emphasize the significance of careful calculation and attention to detail. Mathematical errors can easily creep in, and even a minor misstep can lead to an incorrect solution. By practicing diligence and double-checking our work, we enhance our accuracy and build confidence in our problem-solving abilities. So, let's embark on this mathematical adventure, ready to conquer the equation and discover the hidden value of 't'.

Step 1: Applying the Distributive Property

The first step towards solving the equation 3t - 2(t + 12) = 11 involves simplifying the expression by applying the distributive property. This property allows us to eliminate the parentheses and make the equation more manageable. The distributive property states that a(b + c) = ab + ac. In our case, we have -2 multiplied by the expression (t + 12). Applying the distributive property, we multiply -2 by both 't' and 12. This yields -2 * t = -2t and -2 * 12 = -24. Thus, the term -2(t + 12) becomes -2t - 24.

Rewriting the equation with this simplification, we get 3t - 2t - 24 = 11. This transformation has effectively removed the parentheses, making the equation easier to work with. The distributive property is a fundamental tool in algebra, and mastering its application is crucial for solving various types of equations. It allows us to break down complex expressions into simpler components, paving the way for further simplification and solution. The ability to accurately apply the distributive property is a hallmark of strong algebraic skills.

It's important to pay close attention to the signs when applying the distributive property. In our case, the negative sign in front of the 2 played a crucial role in determining the signs of the resulting terms. Multiplying a negative number by a positive number yields a negative result, while multiplying two negative numbers yields a positive result. Keeping track of these sign rules is essential for avoiding errors and ensuring the accuracy of our calculations. The careful application of the distributive property sets the stage for the next steps in solving the equation, bringing us closer to uncovering the value of 't'. Remember, each step we take builds upon the previous one, highlighting the importance of precision and attention to detail in mathematics.

Step 2: Combining Like Terms

Now that we've applied the distributive property and rewritten the equation as 3t - 2t - 24 = 11, our next step is to combine like terms. Like terms are those that contain the same variable raised to the same power. In this equation, we have two terms containing the variable 't': 3t and -2t. To combine these terms, we simply add their coefficients. The coefficient of 3t is 3, and the coefficient of -2t is -2. Adding these coefficients, we get 3 + (-2) = 1. Therefore, 3t - 2t simplifies to 1t, which is more commonly written as just 't'.

After combining like terms, our equation becomes t - 24 = 11. This simplified equation is much easier to solve than the original one. Combining like terms is a crucial step in simplifying algebraic expressions and equations. It allows us to reduce the number of terms, making the equation more manageable and easier to solve. This process not only simplifies the equation but also helps us to better understand the relationships between the variables and constants involved.

It's important to remember that we can only combine like terms. Terms with different variables or different powers of the same variable cannot be combined. For example, we cannot combine 't' with a constant term like -24. The ability to identify and combine like terms is a fundamental skill in algebra, and it is essential for solving a wide range of equations and problems. By carefully combining like terms, we pave the way for the final step in solving for 't', bringing us closer to our desired solution. This step demonstrates the power of simplification in mathematics, allowing us to tackle complex problems by breaking them down into smaller, more manageable parts.

Step 3: Isolating the Variable 't'

Having simplified the equation to t - 24 = 11, our final step is to isolate the variable 't'. This means we want to get 't' by itself on one side of the equation. To do this, we need to eliminate the -24 term on the left side. We can accomplish this by performing the opposite operation. Since -24 is being subtracted from 't', we add 24 to both sides of the equation. This maintains the equality of the equation, as we are performing the same operation on both sides.

Adding 24 to both sides, we get: t - 24 + 24 = 11 + 24. On the left side, -24 and +24 cancel each other out, leaving us with just 't'. On the right side, 11 + 24 equals 35. Therefore, our equation simplifies to t = 35. This is our solution: the value of 't' that satisfies the original equation.

Isolating the variable is the ultimate goal in solving many algebraic equations. It involves strategically performing operations on both sides of the equation to get the variable by itself. This often requires a combination of addition, subtraction, multiplication, and division, depending on the specific equation. The key is to always perform the opposite operation to eliminate the terms surrounding the variable. In our case, we used addition to eliminate the subtraction of 24. The process of isolating the variable is a fundamental skill in algebra, and it is essential for solving a wide range of problems. By mastering this technique, we empower ourselves to tackle more complex equations and mathematical challenges.

Conclusion: The Value of 't' Revealed

Through a systematic application of algebraic principles, we have successfully solved the equation 3t - 2(t + 12) = 11. Our journey began with simplifying the expression using the distributive property, followed by combining like terms to further refine the equation. Finally, we isolated the variable 't' by strategically adding 24 to both sides of the equation. This meticulous process led us to the solution: t = 35. This final answer represents the value of 't' that makes the original equation true.

The process of solving this equation underscores the importance of several key mathematical concepts. The distributive property allowed us to eliminate parentheses and simplify the expression. Combining like terms helped us to reduce the number of terms and make the equation more manageable. The principle of equality, which dictates that performing the same operation on both sides of an equation maintains its balance, was crucial for isolating the variable. These concepts are fundamental to algebra and are essential for solving a wide variety of mathematical problems. The ability to apply these principles effectively is a testament to strong algebraic skills.

Moreover, this exercise highlights the value of a step-by-step approach to problem-solving. By breaking down the equation into smaller, more manageable steps, we were able to systematically work towards the solution. Each step built upon the previous one, demonstrating the importance of precision and attention to detail. This methodical approach is not only effective in mathematics but also applicable to problem-solving in various other fields. The journey of solving this equation serves as a valuable reminder of the power of perseverance, logical reasoning, and the satisfaction of arriving at a correct solution. The value of 't' is now revealed, a testament to our algebraic prowess and a stepping stone towards tackling more complex mathematical challenges.