Solving For U In -2 = U + 5/9
In the realm of mathematics, solving for unknown variables is a fundamental skill. This article delves into the process of isolating the variable 'u' in the equation -2 = u + 5/9, providing a step-by-step guide to simplify the answer and arrive at the solution. Whether you're a student grappling with algebraic equations or simply seeking to refresh your mathematical prowess, this comprehensive explanation will illuminate the path to solving for 'u'. Equations form the bedrock of various mathematical and scientific disciplines, making the ability to solve them crucial for success in these fields. The principles outlined here can be applied to a wide range of algebraic problems, fostering a deeper understanding of mathematical concepts. So, let's embark on this mathematical journey and unravel the value of 'u'.
Understanding the Equation
The given equation, -2 = u + 5/9, is a linear equation with one variable, 'u'. Our objective is to isolate 'u' on one side of the equation to determine its value. To achieve this, we need to eliminate the term '+ 5/9' from the right side of the equation. The key principle we'll employ is the concept of inverse operations. In this case, the inverse operation of addition is subtraction. Therefore, we will subtract 5/9 from both sides of the equation to maintain the equality. This fundamental principle of maintaining balance is crucial in solving any algebraic equation. By performing the same operation on both sides, we ensure that the equation remains valid and that the value of 'u' is accurately determined. Understanding this core concept is essential for mastering equation-solving techniques.
Step-by-Step Solution
To isolate 'u', we need to subtract 5/9 from both sides of the equation:
-2 - 5/9 = u + 5/9 - 5/9
This step is crucial as it maintains the balance of the equation while moving us closer to isolating 'u'. Now, we need to simplify both sides of the equation. On the right side, +5/9 and -5/9 cancel each other out, leaving us with just 'u'. On the left side, we need to subtract 5/9 from -2. To do this, we first need to express -2 as a fraction with a denominator of 9. This involves multiplying -2 by 9/9, which gives us -18/9. The equation now becomes:
-18/9 - 5/9 = u
Now, we can subtract the fractions on the left side. Since they have the same denominator, we simply subtract the numerators:
(-18 - 5)/9 = u
-23/9 = u
Therefore, the value of 'u' is -23/9. This fraction is already in its simplest form as 23 and 9 have no common factors other than 1.
Expressing the Solution
The solution u = -23/9 can be expressed in different forms, depending on the context and desired level of precision. It is currently in the form of an improper fraction, where the numerator is greater than the denominator. This form is mathematically accurate and often preferred in algebraic contexts. However, we can also express it as a mixed number, which combines a whole number and a proper fraction. To do this, we divide -23 by 9. The quotient is -2, and the remainder is -5. Therefore, the mixed number representation is -2 5/9. This form can be more intuitive for some, as it clearly shows the whole number part and the fractional part. Alternatively, we can express the solution as a decimal by dividing -23 by 9 using a calculator or long division. This yields an approximate decimal value of -2.555..., where the 5s repeat infinitely. We can round this decimal to a desired level of precision, such as -2.56 or -2.556, depending on the required accuracy.
Verification
To ensure the accuracy of our solution, it's crucial to verify it by substituting the value of u = -23/9 back into the original equation:
-2 = u + 5/9
Substituting u = -23/9, we get:
-2 = -23/9 + 5/9
Now, we need to simplify the right side of the equation. Since the fractions have the same denominator, we can add the numerators:
-2 = (-23 + 5)/9
-2 = -18/9
Simplifying the fraction -18/9, we get:
-2 = -2
Since both sides of the equation are equal, our solution u = -23/9 is correct. This verification step is a crucial part of the problem-solving process, as it helps to identify any potential errors and ensures that the final answer is accurate. By substituting the solution back into the original equation, we can confidently confirm its validity.
Conclusion
In this article, we have successfully solved for 'u' in the equation -2 = u + 5/9. By employing the principle of inverse operations and maintaining the balance of the equation, we arrived at the solution u = -23/9. We further explored different ways to express this solution, including as an improper fraction, a mixed number, and a decimal approximation. The importance of verification was also emphasized, as it provides a crucial check for the accuracy of our solution. Mastering the techniques presented in this article is essential for tackling more complex algebraic problems and building a strong foundation in mathematics. The ability to solve equations is a fundamental skill that extends beyond the classroom, finding applications in various fields such as science, engineering, and finance. By understanding the underlying principles and practicing regularly, you can enhance your problem-solving abilities and confidently navigate the world of equations. Remember, practice makes perfect, so continue to explore and challenge yourself with new mathematical problems.
