Solving For Y A Step By Step Guide To Linear Equations
Linear equations are the foundation of algebra, and mastering their solution is crucial for success in higher mathematics. In this comprehensive guide, we will delve into the intricacies of solving for a variable, specifically 'y,' in a linear equation. We will break down the process step by step, ensuring a clear understanding of each operation and its purpose. By the end of this guide, you will be equipped with the skills and knowledge to confidently tackle any linear equation that comes your way. Linear equations are algebraic expressions where the highest power of the variable is 1. They can be written in the general form ax + b = c, where a, b, and c are constants, and x is the variable we want to solve for. Solving a linear equation involves isolating the variable on one side of the equation by performing a series of operations on both sides to maintain the equality. The goal is to manipulate the equation until the variable stands alone, revealing its value. This process often involves simplifying the equation by combining like terms, distributing constants, and using inverse operations to undo addition, subtraction, multiplication, and division. Understanding these fundamental principles is key to successfully navigating the world of linear equations. Mastering these techniques not only builds a strong foundation in algebra but also enhances problem-solving skills applicable in various real-world scenarios. Whether you're a student grappling with homework or an individual seeking to sharpen your mathematical abilities, this guide provides a comprehensive resource for conquering linear equations.
The Given Equation: A Detailed Examination
Let's begin by carefully examining the equation we aim to solve:
-2y - 8(-5y - 1) = -5 + 2(7 - y)
This equation presents a classic example of a linear equation with the variable 'y.' Our objective is to isolate 'y' on one side of the equation to determine its value. To achieve this, we will employ a series of algebraic manipulations, ensuring that each operation maintains the equation's balance. The equation's structure reveals several key elements that require our attention. First, we observe the presence of parentheses, which indicate the need for the distributive property. This property allows us to multiply the constant outside the parentheses by each term inside, effectively expanding the expression. Second, we notice the presence of like terms, which are terms that contain the same variable raised to the same power. Combining like terms simplifies the equation and makes it easier to isolate the variable. Third, we encounter constants on both sides of the equation, which will need to be strategically moved to consolidate them on one side. The negative signs and coefficients also play a crucial role in the equation's solution. Paying close attention to these details is essential to avoid errors and arrive at the correct answer. Before we dive into the step-by-step solution, it's important to emphasize the significance of maintaining balance throughout the process. Any operation performed on one side of the equation must be mirrored on the other side to preserve the equality. This principle forms the cornerstone of solving equations and ensures the validity of our solution. In the following sections, we will meticulously dissect the equation, applying the necessary operations and simplifying steps to unveil the value of 'y'.
Step 1: Applying the Distributive Property
The first step in solving this equation involves applying the distributive property to eliminate the parentheses. This property states that a(b + c) = ab + ac. We will apply this property to both sides of the equation:
-2y - 8(-5y - 1) = -5 + 2(7 - y)
On the left side, we distribute the -8:
-2y + 40y + 8 = -5 + 2(7 - y)
Remember that multiplying a negative number by a negative number results in a positive number. So, -8 * -5y becomes +40y, and -8 * -1 becomes +8. On the right side, we distribute the 2:
-2y + 40y + 8 = -5 + 14 - 2y
Here, 2 * 7 equals 14, and 2 * -y equals -2y. The distributive property is a fundamental tool in simplifying algebraic expressions. It allows us to expand expressions and remove parentheses, making the equation easier to manipulate. By carefully applying this property, we've transformed the original equation into a more manageable form. The next step will involve combining like terms on each side of the equation. This process further simplifies the equation and brings us closer to isolating the variable 'y'. It's crucial to pay attention to the signs of the terms and perform the operations accurately to avoid errors. The distributive property is not just a mathematical trick; it's a logical consequence of how multiplication interacts with addition and subtraction. Understanding this principle helps in grasping the underlying structure of algebraic expressions and builds a stronger foundation for more advanced mathematical concepts. In the subsequent steps, we will continue to employ various algebraic techniques to isolate 'y' and determine its value. Each step is carefully designed to maintain the equation's balance and bring us closer to the solution. The key is to approach the problem systematically and apply the rules of algebra with precision.
Step 2: Combining Like Terms
Now that we've applied the distributive property, our equation looks like this:
-2y + 40y + 8 = -5 + 14 - 2y
The next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. On the left side, we have -2y and +40y, which can be combined:
38y + 8 = -5 + 14 - 2y
-2y + 40y equals 38y. On the right side, we have -5 and +14, which are constants and can be combined:
38y + 8 = 9 - 2y
-5 + 14 equals 9. Combining like terms is a crucial step in simplifying equations. It reduces the number of terms and makes the equation easier to solve. By grouping similar terms together, we streamline the equation and bring it closer to a form where the variable can be isolated. This process highlights the importance of identifying and understanding the different types of terms in an equation. Variables and constants behave differently, and combining them requires careful consideration. In this step, we've effectively consolidated the terms on each side of the equation, setting the stage for the next phase of the solution. The equation now presents a clearer picture of the relationship between the variable 'y' and the constants. As we move forward, we will continue to manipulate the equation, ensuring that each step maintains the balance and brings us closer to the final answer. Combining like terms is not just a mechanical process; it's a strategic step that simplifies the equation and makes it more manageable. By understanding the logic behind this step, we can approach equation solving with greater confidence and clarity. The ability to identify and combine like terms is a fundamental skill in algebra and serves as a building block for more complex mathematical concepts. In the upcoming steps, we will employ additional algebraic techniques to isolate the variable and determine its value.
Step 3: Isolating the Variable Term
Our equation currently stands as:
38y + 8 = 9 - 2y
To isolate the variable term, we need to get all the 'y' terms on one side of the equation. Let's add 2y to both sides:
38y + 8 + 2y = 9 - 2y + 2y
This gives us:
40y + 8 = 9
Adding 2y to both sides cancels out the -2y on the right side and combines with the 38y on the left side. The principle behind isolating the variable term is to strategically eliminate the variable from one side of the equation. By performing the same operation on both sides, we maintain the equation's balance while moving the variable terms to a single location. This step is crucial because it simplifies the equation and brings us closer to isolating the variable itself. The choice of adding 2y to both sides is driven by the desire to eliminate the -2y term on the right side. This is an example of using inverse operations to undo the effect of a term. Adding 2y is the inverse operation of subtracting 2y, and when these operations are applied together, they cancel each other out. Isolating the variable term is not just a procedural step; it's a strategic move that simplifies the equation and makes it easier to solve. By understanding the logic behind this step, we can approach equation solving with greater confidence and clarity. The ability to isolate the variable term is a fundamental skill in algebra and serves as a building block for more complex mathematical concepts. In the upcoming steps, we will continue to manipulate the equation, ensuring that each step maintains the balance and brings us closer to the final answer. The next step will involve isolating the variable itself by removing the constant term from the side with the variable.
Step 4: Isolating the Variable
Now we have:
40y + 8 = 9
To isolate 'y', we need to get rid of the +8 on the left side. We can do this by subtracting 8 from both sides:
40y + 8 - 8 = 9 - 8
This simplifies to:
40y = 1
Subtracting 8 from both sides isolates the term with 'y' on the left side. This step demonstrates the power of inverse operations in solving equations. Subtraction is the inverse operation of addition, and by subtracting 8 from both sides, we effectively undo the addition of 8 on the left side. This process leaves us with the variable term alone, making it easier to isolate the variable itself. Isolating the variable is a critical step in solving equations. It involves removing all the terms and constants that are attached to the variable, leaving it standing alone on one side of the equation. This process requires a careful application of inverse operations, ensuring that each operation is performed on both sides of the equation to maintain balance. The choice of subtracting 8 from both sides is driven by the desire to eliminate the +8 term on the left side. This is a strategic decision that simplifies the equation and brings us closer to the final answer. As we move forward, we will continue to manipulate the equation, ensuring that each step maintains the balance and brings us closer to the solution. The next step will involve dividing both sides by the coefficient of 'y' to completely isolate the variable. The ability to isolate the variable is a fundamental skill in algebra and serves as a building block for more complex mathematical concepts. By understanding the logic behind this step, we can approach equation solving with greater confidence and clarity. In the upcoming steps, we will employ the final algebraic technique to unveil the value of 'y'.
Step 5: Solving for y
We're almost there! Our equation is now:
40y = 1
To solve for 'y', we need to divide both sides by 40:
40y / 40 = 1 / 40
This gives us the solution:
y = 1/40
Dividing both sides by 40 isolates 'y' and provides the solution to the equation. This final step demonstrates the power of division as the inverse operation of multiplication. By dividing both sides by the coefficient of 'y', we effectively undo the multiplication and reveal the value of 'y'. Solving for the variable is the ultimate goal of equation solving. It involves performing the necessary operations to isolate the variable and determine its numerical value. This process requires a careful application of inverse operations, ensuring that each operation is performed on both sides of the equation to maintain balance. The choice of dividing both sides by 40 is driven by the desire to eliminate the coefficient of 'y'. This is a strategic decision that isolates the variable and provides the solution to the equation. We have successfully solved for 'y' and found its value to be 1/40. This process highlights the importance of understanding the properties of equality and the use of inverse operations in solving equations. The ability to solve for a variable is a fundamental skill in algebra and serves as a building block for more complex mathematical concepts. By understanding the logic behind this step, we can approach equation solving with greater confidence and clarity. In the next section, we will verify our solution to ensure its accuracy.
Solution: y = 1/40
Therefore, the solution to the equation is:
y = 1/40
Conclusion: Mastering Linear Equations
In this comprehensive guide, we have meticulously walked through the process of solving a linear equation for 'y.' We began by examining the equation, identifying the key elements and operations required. We then systematically applied the distributive property, combined like terms, and isolated the variable term. Finally, we solved for 'y' by dividing both sides of the equation by its coefficient. Throughout the process, we emphasized the importance of maintaining balance by performing the same operations on both sides of the equation. This principle is fundamental to solving equations and ensures the accuracy of the solution. Mastering linear equations is a crucial step in developing a strong foundation in algebra. These equations form the basis for more advanced mathematical concepts and are widely applicable in various real-world scenarios. By understanding the techniques and strategies presented in this guide, you can confidently tackle linear equations and unlock a deeper understanding of mathematics. The ability to solve linear equations is not just a mathematical skill; it's a problem-solving tool that can be applied in diverse fields, from science and engineering to finance and economics. The systematic approach we have employed in this guide can be adapted to solve a wide range of mathematical problems. The key is to break down complex problems into smaller, manageable steps and apply the appropriate techniques with precision. We encourage you to practice solving linear equations regularly to reinforce your understanding and build your confidence. The more you practice, the more proficient you will become in applying the techniques and strategies presented in this guide. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively to solve problems. With dedication and practice, you can master linear equations and unlock the beauty and power of mathematics.
