Solving For T In The Equation (-5/6) = (1/(t-4)) A Step-by-Step Guide
In the realm of mathematics, solving for a specific variable within an equation is a fundamental skill. This comprehensive guide will walk you through the process of isolating 't' in the equation (-5/6) = (1/(t-4)). Understanding how to manipulate equations and solve for unknowns is crucial for various applications, from basic algebra to advanced calculus and real-world problem-solving. This article provides a detailed, step-by-step approach to tackle this specific equation, ensuring clarity and a solid grasp of the underlying principles. We will explore the concepts of cross-multiplication, algebraic manipulation, and the importance of maintaining balance in equations. By the end of this guide, you will not only be able to solve this particular equation but also gain a deeper understanding of the techniques involved in solving similar problems. This knowledge will empower you to confidently approach a wide range of mathematical challenges, enhancing your problem-solving abilities and analytical thinking skills. Mastering these techniques is essential for anyone pursuing further studies in mathematics, science, engineering, or any field that requires quantitative reasoning. So, let's embark on this journey of mathematical exploration and unlock the secrets to solving equations effectively and efficiently.
Understanding the Equation Structure
Before diving into the solution, let's analyze the structure of the equation:
(-5/6) = (1/(t-4))
This equation represents a proportion, where two fractions are set equal to each other. The variable 't' is located in the denominator of the fraction on the right side of the equation. To isolate 't', we need to eliminate the fractions and bring 't' to the numerator. This involves a series of algebraic manipulations, which we will explore in detail. The key to success lies in understanding the properties of equality, which allow us to perform operations on both sides of the equation without changing its fundamental truth. This principle of balance is crucial in solving any equation, ensuring that we maintain the relationship between the two sides. By carefully applying these principles, we can systematically isolate the variable 't' and arrive at the correct solution. This initial analysis sets the stage for the subsequent steps, providing a clear roadmap for our problem-solving journey. Let's move on to the next step and begin the process of manipulating the equation to isolate 't'.
Step 1: Cross-Multiplication
The first step to solving this equation is to eliminate the fractions. We can achieve this by using the method of cross-multiplication. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction and vice versa. In our equation:
(-5/6) = (1/(t-4))
We cross-multiply as follows:
-5 * (t - 4) = 1 * 6
This step effectively removes the fractions, transforming the equation into a more manageable form. The concept of cross-multiplication is based on the fundamental property of proportions, which states that if two ratios are equal, then their cross-products are also equal. This principle allows us to convert a proportional equation into a linear equation, making it easier to solve for the unknown variable. By performing this operation, we have simplified the equation and paved the way for further algebraic manipulations. The resulting equation, -5 * (t - 4) = 1 * 6, is now free of fractions and can be solved using standard algebraic techniques. This step is crucial in the overall solution process, as it lays the foundation for isolating the variable 't' and determining its value. Let's proceed to the next step, where we will further simplify the equation and continue our journey towards the solution.
Step 2: Distribute and Simplify
Now, let's distribute the -5 on the left side of the equation:
-5 * (t - 4) = 1 * 6
This means we multiply -5 by both 't' and -4:
-5t + 20 = 6
This step is crucial in simplifying the equation and bringing us closer to isolating the variable 't'. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a group of terms within parentheses. By applying this property correctly, we can expand the equation and eliminate the parentheses, making it easier to manipulate and solve. In this case, distributing the -5 results in the equation -5t + 20 = 6, which is a linear equation in one variable. This form is much simpler to work with compared to the original equation, which involved fractions and parentheses. Simplifying equations is a key skill in algebra, as it allows us to transform complex expressions into simpler forms that are easier to understand and solve. By mastering this skill, you will be able to tackle a wide range of mathematical problems with confidence and efficiency. Let's move on to the next step, where we will continue to isolate 't' and ultimately find its value.
Step 3: Isolate the Term with 't'
To isolate the term with 't', we need to subtract 20 from both sides of the equation:
-5t + 20 = 6
Subtracting 20 from both sides gives us:
-5t + 20 - 20 = 6 - 20
-5t = -14
This step is a crucial part of the process of isolating the variable 't'. By subtracting 20 from both sides of the equation, we maintain the balance of the equation while moving closer to our goal of having 't' by itself on one side. This operation is based on the fundamental property of equality, which states that we can perform the same operation on both sides of an equation without changing its solution. Isolating the term with 't' is a key strategy in solving algebraic equations, as it allows us to focus on the variable we are trying to find. By carefully applying this technique, we can systematically eliminate the other terms in the equation and eventually determine the value of 't'. The resulting equation, -5t = -14, is now much simpler than the original equation and can be easily solved in the next step. Let's proceed to the final step, where we will divide both sides by -5 to solve for 't'.
Step 4: Solve for 't'
Finally, to solve for 't', we need to divide both sides of the equation by -5:
-5t = -14
Dividing both sides by -5, we get:
t = (-14) / (-5)
t = 14/5
Therefore, the solution for 't' is 14/5.
This final step completes the process of solving for 't' in the given equation. By dividing both sides of the equation by -5, we have successfully isolated 't' and determined its value. This operation is based on the same fundamental property of equality that we used in the previous steps, ensuring that the equation remains balanced and the solution is correct. The result, t = 14/5, is the value that satisfies the original equation. This means that if we substitute 14/5 for 't' in the equation (-5/6) = (1/(t-4)), both sides of the equation will be equal. Solving for variables is a fundamental skill in algebra, and this example demonstrates the step-by-step process of isolating a variable and finding its value. By mastering this technique, you will be able to solve a wide range of algebraic equations and apply this knowledge to various mathematical and real-world problems. Congratulations on successfully solving for 't'!
Conclusion
In this guide, we have thoroughly walked through the process of solving for 't' in the equation (-5/6) = (1/(t-4)). We started by understanding the equation's structure, then applied cross-multiplication to eliminate fractions, distributed and simplified the equation, isolated the term with 't', and finally, solved for 't'. The solution we found is t = 14/5. This step-by-step approach demonstrates the importance of algebraic manipulation and the application of fundamental mathematical principles. Solving equations is a core skill in mathematics, and the techniques discussed here are applicable to a wide range of problems. By mastering these techniques, you can enhance your problem-solving abilities and gain a deeper understanding of mathematical concepts. This guide serves as a valuable resource for anyone looking to improve their algebraic skills and confidently tackle equations. Remember, practice is key to mastering these concepts, so be sure to apply these techniques to other problems and continue to expand your mathematical knowledge. The journey of mathematical exploration is a rewarding one, and with each problem solved, you build a stronger foundation for future learning and success.