Solving For Y In The Equation 30 - 28 = 30y - 14y - 7y A Step-by-Step Guide

Introduction

In this article, we will walk through the process of solving for the variable y in the algebraic equation 30 - 28 = 30y - 14y - 7y. Solving algebraic equations is a fundamental skill in mathematics, and it involves isolating the variable on one side of the equation to determine its value. This process typically includes simplifying the equation by combining like terms and performing inverse operations to maintain the equality. Understanding how to solve such equations is crucial for various mathematical applications, including algebra, calculus, and other advanced topics. This article aims to provide a comprehensive, step-by-step solution, making it easy for students and anyone interested in mathematics to grasp the method and apply it to similar problems. We will break down each step with clear explanations, ensuring that the underlying logic is well understood. Mastering these skills builds a solid foundation for tackling more complex mathematical problems. Therefore, let's dive into solving this equation and reinforce our understanding of algebraic manipulations.

Step-by-Step Solution

To solve the equation 30 - 28 = 30y - 14y - 7y, we need to follow a systematic approach that involves simplifying both sides of the equation and isolating the variable y. Here is a detailed, step-by-step solution:

1. Simplify Both Sides of the Equation

The first step in solving any algebraic equation is to simplify both sides as much as possible. This involves performing any arithmetic operations and combining like terms. On the left side of the equation, we have a simple subtraction: 30 - 28. Performing this subtraction gives us 2. So, the left side of the equation simplifies to 2. On the right side of the equation, we have terms involving the variable y. Specifically, we have 30y, -14y, and -7y. These are like terms because they all contain the same variable, y, raised to the same power (which is 1 in this case). To combine these like terms, we add their coefficients. The coefficients are the numbers multiplied by y, which are 30, -14, and -7. Adding these coefficients, we get 30 + (-14) + (-7). First, let's add 30 and -14, which gives us 16. Then, we add 16 and -7, which gives us 9. So, the right side of the equation simplifies to 9y. Now, our equation looks much simpler: 2 = 9y. This simplified form makes it easier to isolate the variable and find its value. Simplifying both sides of an equation is crucial because it reduces the complexity and makes the subsequent steps more straightforward. By performing the arithmetic operations and combining like terms, we have transformed the original equation into a more manageable form, setting the stage for solving for y.

2. Isolate the Variable y

After simplifying both sides of the equation, our next goal is to isolate the variable y. This means getting y by itself on one side of the equation. Currently, we have the equation 2 = 9y. The variable y is being multiplied by 9. To isolate y, we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by the coefficient of y, which is 9. This ensures that we maintain the equality of the equation. Dividing both sides by 9, we get 2/9 = (9y)/9. On the right side, the 9 in the numerator and the 9 in the denominator cancel each other out, leaving us with just y. So, the equation becomes 2/9 = y. This tells us that the value of y is 2/9. Isolating the variable is a fundamental step in solving algebraic equations. By performing the correct inverse operation, we can undo the operation that is being applied to the variable. In this case, we divided both sides by 9 to undo the multiplication. This process is essential for solving various types of equations, including linear equations, quadratic equations, and systems of equations. The ability to isolate variables allows us to determine their values and understand the relationships between different quantities in mathematical problems. Now that we have found the value of y, we can move on to verifying our solution to ensure it is correct.

3. Verify the Solution

To ensure that our solution is correct, we need to verify it by substituting the value we found for y back into the original equation. Our solution is y = 2/9. The original equation is 30 - 28 = 30y - 14y - 7y. We will substitute 2/9 for y in the equation and check if both sides of the equation are equal. Substituting y = 2/9 into the right side of the equation, we get 30*(2/9) - 14*(2/9) - 7*(2/9). To simplify this, we first multiply each term by 2/9. 30*(2/9) is equal to 60/9. 14*(2/9) is equal to 28/9. 7*(2/9) is equal to 14/9. So, the expression becomes 60/9 - 28/9 - 14/9. Now, we subtract the fractions. Since they all have the same denominator, we can subtract the numerators. 60 - 28 - 14 gives us 18. So, the expression simplifies to 18/9. We can further simplify 18/9 by dividing both the numerator and the denominator by 9. This gives us 2. Now, let's look at the left side of the original equation, which is 30 - 28. This simplifies to 2. We found that the right side of the equation also simplifies to 2 when we substitute y = 2/9. Since both sides of the equation are equal (2 = 2), our solution y = 2/9 is correct. Verifying the solution is an important step in the problem-solving process. It helps us catch any errors we might have made during the simplification or isolation steps. By substituting the solution back into the original equation, we can confirm that our answer satisfies the equation and is indeed the correct value for the variable. This step provides confidence in our solution and reinforces our understanding of the problem.

Final Answer

After simplifying the equation, isolating the variable, and verifying the solution, we have confidently determined the value of y. The final answer is y = 2/9. This means that when y is equal to 2/9, the equation 30 - 28 = 30y - 14y - 7y holds true. This result can be used in further mathematical calculations or applications where this equation is relevant. Understanding how to solve for variables in algebraic equations is a fundamental skill that is used throughout mathematics and various scientific disciplines. The ability to manipulate equations and find solutions is essential for problem-solving and analytical thinking. This step-by-step guide has provided a clear and thorough method for solving this particular equation, and the same principles can be applied to a wide range of similar problems. The final answer, y = 2/9, represents the value that satisfies the given equation and completes the solution.

In conclusion, we have successfully solved for y in the equation 30 - 28 = 30y - 14y - 7y. We achieved this by following a systematic approach that included simplifying both sides of the equation, combining like terms, isolating the variable, and verifying our solution. The step-by-step process allowed us to break down the problem into manageable parts, making it easier to understand and solve. First, we simplified both sides of the equation. On the left side, 30 - 28 simplified to 2. On the right side, we combined the like terms 30y, -14y, and -7y to get 9y. This gave us the simplified equation 2 = 9y. Next, we isolated the variable y by dividing both sides of the equation by 9. This resulted in y = 2/9. To ensure the accuracy of our solution, we verified it by substituting y = 2/9 back into the original equation. After performing the necessary calculations, we confirmed that both sides of the equation were equal, thus validating our solution. The final answer, y = 2/9, is the value that makes the equation true. This exercise demonstrates the importance of following a structured approach when solving algebraic equations. Simplifying, isolating, and verifying are key steps that help ensure the correctness of the solution. The skills learned in solving this equation can be applied to a wide variety of mathematical problems, making it a valuable learning experience. By mastering these techniques, students and math enthusiasts can confidently tackle more complex equations and further advance their mathematical abilities. The process of solving for variables is a fundamental aspect of algebra and is crucial for success in higher-level mathematics and related fields.