Solving The Equation (5-6y)/3 + Y/8 = 0 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving a specific linear equation: $\frac{5-6y}{3} + \frac{y}{8} = 0$. We will break down each step in detail, ensuring clarity and understanding for readers of all backgrounds. This comprehensive guide will not only provide the solution but also elucidate the underlying principles, making it a valuable resource for anyone seeking to enhance their algebraic proficiency. Understanding how to solve linear equations is crucial for various applications in science, engineering, and everyday problem-solving. So, let's embark on this mathematical journey and conquer this equation together.
Understanding the Equation
Before we dive into the solution, let's first understand the equation we are dealing with: $\frac{5-6y}{3} + \frac{y}{8} = 0$. This is a linear equation in one variable, y. Linear equations are characterized by having the variable raised to the power of 1, and they can be represented graphically as a straight line. Our goal is to find the value of y that satisfies this equation, meaning the value that makes the left-hand side equal to the right-hand side (which is 0 in this case). The equation involves fractions, which adds a layer of complexity. To solve it effectively, we will need to employ techniques such as finding a common denominator and simplifying expressions. Recognizing the structure of the equation is the first step towards finding its solution. We will use algebraic manipulations to isolate y on one side of the equation, ultimately revealing its value. This process will involve several key steps, each of which we will explore in detail to ensure a thorough understanding.
Step 1: Eliminating Fractions
To simplify the equation $\frac5-6y}{3} + \frac{y}{8} = 0$, the first crucial step is to eliminate the fractions. Fractions can make equations cumbersome to work with, so removing them often makes the process smoother. To do this, we need to find the least common multiple (LCM) of the denominators, which are 3 and 8 in this case. The LCM of 3 and 8 is 24. Now, we multiply both sides of the equation by this LCM. Multiplying both sides of an equation by the same non-zero value maintains the equality, a fundamental principle in algebra. This gives us3} + \frac{y}{8}) = 24 * 0$. Distributing the 24 on the left side, we get3} + 24 * \frac{y}{8} = 0$. Now we can simplify each term{3} * (5-6y) + \frac{24}{8} * y = 0$. This simplifies to: $8 * (5-6y) + 3 * y = 0$. By eliminating the fractions, we have transformed the equation into a more manageable form, setting the stage for the next steps in solving for y. This step is a common strategy in solving equations involving fractions and is essential for efficient manipulation.
Step 2: Distributing and Simplifying
Following the elimination of fractions, our equation now stands as: $8 * (5-6y) + 3 * y = 0$. The next step involves distributing the 8 across the terms inside the parentheses. This means multiplying 8 by both 5 and -6y. Performing this multiplication, we get: $8 * 5 - 8 * 6y + 3y = 0$, which simplifies to: $40 - 48y + 3y = 0$. Now, we have an equation with constant terms and terms involving y. The next part of this step is to simplify the equation by combining like terms. In this case, we can combine the -48y and +3y terms. This gives us: $40 + (-48y + 3y) = 0$, which further simplifies to: $40 - 45y = 0$. By distributing and simplifying, we have reduced the equation to a more concise form, making it easier to isolate the variable y. This step is a crucial part of the algebraic manipulation process, allowing us to move closer to the solution.
Step 3: Isolating the Variable
Having simplified the equation to $40 - 45y = 0$, our next objective is to isolate the variable y. This means getting y by itself on one side of the equation. To begin, we need to eliminate the constant term, which is 40 in this case. We can do this by subtracting 40 from both sides of the equation. This maintains the equality while moving the constant term to the right side. Subtracting 40 from both sides, we get: $40 - 45y - 40 = 0 - 40$, which simplifies to: $-45y = -40$. Now, the term with y is isolated on the left side. To completely isolate y, we need to get rid of the coefficient -45. This can be achieved by dividing both sides of the equation by -45. Dividing both sides by -45, we get: $\frac-45y}{-45} = \frac{-40}{-45}$. This simplifies to{45}$. We have now successfully isolated y, but the fraction can be further simplified. This isolation process is a fundamental technique in solving equations, allowing us to determine the value of the unknown variable.
Step 4: Simplifying the Solution
After isolating y, we arrived at the solution $y = \frac40}{45}$. However, this fraction is not in its simplest form. To simplify it, we need to find the greatest common divisor (GCD) of the numerator (40) and the denominator (45). The GCD is the largest number that divides both 40 and 45 without leaving a remainder. In this case, the GCD of 40 and 45 is 5. To simplify the fraction, we divide both the numerator and the denominator by their GCD. Dividing both 40 and 45 by 5, we get{45 \div 5} = \frac{8}{9}$. Therefore, the simplified solution for y is $\frac{8}{9}$. This means that the value of y that satisfies the original equation is 8/9. Simplifying the solution is an important step in presenting the answer in its most concise and understandable form. It also helps in recognizing the solution more easily in different contexts.
Final Solution
After meticulously following each step, we have successfully solved the equation $\frac5-6y}{3} + \frac{y}{8} = 0$. The final solution, after simplifying, is{9}$. This means that when y is equal to 8/9, the left-hand side of the original equation will equal zero, thus satisfying the equation. To recap, we started by eliminating fractions by multiplying both sides of the equation by the least common multiple of the denominators. Then, we distributed and simplified the equation, followed by isolating the variable y. Finally, we simplified the resulting fraction to obtain the most concise form of the solution. This process demonstrates the power of algebraic manipulation in solving linear equations. Understanding these steps not only allows you to solve this particular equation but also equips you with the skills to tackle a wide range of similar problems. The journey to solving this equation highlights the importance of precision and attention to detail in mathematics. With a solid grasp of these techniques, you can confidently approach and solve various algebraic equations.
