Solving For Y In The Equation 4x - 3y = 11 When X Equals 2
Introduction
In the realm of mathematics, solving equations is a fundamental skill. We often encounter linear equations in two variables, which represent relationships between two unknown quantities. One such equation is 4x - 3y = 11. This article delves into the process of solving this equation to determine the value of 'y' when 'x' is given as 2. This is a crucial concept in algebra, with applications spanning various fields, including physics, engineering, and economics. By understanding how to solve such equations, we gain the ability to model and analyze real-world scenarios, making informed decisions based on mathematical principles. Understanding the principles behind solving such equations is essential for anyone pursuing studies or careers involving quantitative analysis. This article provides a step-by-step guide, ensuring clarity and comprehension for learners of all levels. We will explore the underlying logic and techniques involved, offering a solid foundation for tackling more complex algebraic problems. Solving linear equations is not just an academic exercise; it's a practical skill that empowers us to solve problems in various aspects of life. So, let's embark on this journey to unravel the solution to this equation and enhance our mathematical prowess.
Understanding Linear Equations
Before we dive into the solution, let's clarify what a linear equation is. A linear equation in two variables (typically x and y) can be written in the form Ax + By = C, where A, B, and C are constants. In our case, the equation 4x - 3y = 11 fits this form, with A = 4, B = -3, and C = 11. Linear equations represent straight lines when graphed on a coordinate plane. Each point on the line represents a solution to the equation, meaning the x and y coordinates of the point satisfy the equation. The concept of linearity is crucial in mathematics, as it allows for straightforward analysis and prediction. Linear relationships are often used to approximate real-world phenomena, providing a simplified yet accurate model for understanding complex systems. Understanding the properties of linear equations is essential for solving them effectively. This includes recognizing the coefficients and constants, understanding the concept of slope and intercept, and being able to manipulate the equation to isolate the desired variable. Linear equations form the backbone of many mathematical models, making their comprehension paramount for anyone seeking to apply mathematics in practical scenarios. The ability to solve linear equations efficiently and accurately is a valuable asset in various fields, ranging from science and engineering to finance and economics. So, a solid grasp of the fundamentals of linear equations is the first step towards mastering more advanced mathematical concepts.
Step-by-Step Solution
Now, let's walk through the steps to solve the equation 4x - 3y = 11 when x = 2. This process involves substituting the given value of x into the equation and then solving for y. This is a common technique in algebra, used to find the value of one variable when the other is known. The key is to follow the order of operations and maintain the balance of the equation. Each step must be performed carefully to avoid errors and ensure an accurate result. Understanding the rationale behind each step is crucial for developing problem-solving skills and applying these techniques to other equations. The ability to solve for a specific variable is a fundamental skill in algebra, with applications in various mathematical contexts. From graphing lines to solving systems of equations, this technique is a cornerstone of algebraic manipulation. So, let's break down the process into manageable steps and gain a clear understanding of how to arrive at the solution.
Step 1: Substitute the Value of x
Substitute x = 2 into the equation 4x - 3y = 11. This gives us 4(2) - 3y = 11. The substitution step is crucial because it allows us to reduce the equation from two variables to one. By replacing 'x' with its numerical value, we create an equation that only involves 'y', making it solvable. This technique is widely used in algebra to simplify equations and isolate the unknown variable. The accuracy of the substitution is paramount, as any error in this step will propagate through the rest of the solution. It's important to pay close attention to the signs and coefficients when performing the substitution. This step is not just about replacing a variable with a number; it's about transforming the equation into a more manageable form. By understanding the logic behind substitution, we can apply this technique to a wide range of algebraic problems. So, with the value of 'x' substituted, we can now proceed to simplify the equation and solve for 'y'.
Step 2: Simplify the Equation
Simplify the equation 4(2) - 3y = 11. This becomes 8 - 3y = 11. The simplification process involves performing the arithmetic operations to reduce the equation to its simplest form. In this case, we multiply 4 by 2 to get 8. This step is crucial for making the equation easier to manipulate and solve. By simplifying the equation, we reduce the number of terms and operations, making it clearer what steps need to be taken next. Simplification often involves combining like terms, distributing coefficients, and performing basic arithmetic. The goal is to isolate the variable we're trying to solve for, in this case, 'y'. Accurate simplification is essential for obtaining the correct solution. Any errors in this step can lead to an incorrect answer. Therefore, it's important to double-check the calculations and ensure that each operation is performed correctly. With the equation simplified to 8 - 3y = 11, we're one step closer to isolating 'y' and finding its value.
Step 3: Isolate the Term with y
To isolate the term with y, subtract 8 from both sides of the equation 8 - 3y = 11. This gives us -3y = 11 - 8, which simplifies to -3y = 3. The isolation of the variable is a key step in solving any equation. By performing the same operation on both sides of the equation, we maintain the equality while moving terms around. In this case, we subtract 8 from both sides to get the term with 'y' alone on one side. This process is based on the fundamental principle of algebraic manipulation: maintaining balance. Any operation performed on one side of the equation must be performed on the other side to ensure that the equation remains true. Isolating the variable is often the most challenging part of solving equations, but it's also the most crucial. Once the variable is isolated, the remaining steps are usually straightforward. With -3y = 3, we're now ready to solve for 'y' directly.
Step 4: Solve for y
Divide both sides of the equation -3y = 3 by -3 to solve for y. This results in y = 3 / -3, which simplifies to y = -1. The final step in solving for 'y' is to divide both sides of the equation by the coefficient of 'y'. This isolates 'y' and gives us its value. In this case, we divide both sides by -3, resulting in y = -1. This step demonstrates the inverse operation principle: to undo multiplication, we divide. This principle is fundamental to solving algebraic equations. It's important to pay attention to the signs when dividing, as a negative sign can change the result. With this step, we have successfully solved the equation for 'y' when x = 2. The solution y = -1 satisfies the original equation 4x - 3y = 11. This value represents the point on the line represented by the equation where the x-coordinate is 2. So, with the value of 'y' determined, we have completed the solution process.
Final Answer
Therefore, when x = 2, the value of y in the equation 4x - 3y = 11 is -1. The final answer is the culmination of all the steps we've taken. It represents the solution to the problem, the value of 'y' that satisfies the equation when x = 2. This answer is not just a number; it's a point on the line represented by the equation. It's the intersection of the line with the vertical line x = 2. Presenting the final answer clearly and concisely is crucial for effective communication of the solution. It's also important to double-check the answer by substituting it back into the original equation to ensure that it satisfies the equation. This verification step helps to prevent errors and build confidence in the solution. With the final answer determined, we have successfully solved the problem and demonstrated our understanding of linear equations.
Verification
To verify our solution, substitute x = 2 and y = -1 back into the original equation 4x - 3y = 11. This gives us 4(2) - 3(-1) = 11. Simplifying, we get 8 + 3 = 11, which is true. Verification is a crucial step in the problem-solving process. It ensures that our solution is correct and that we haven't made any errors along the way. By substituting the values of 'x' and 'y' back into the original equation, we can check if the equation holds true. If the equation is satisfied, then our solution is correct. If not, then we need to go back and review our steps to find the error. Verification not only confirms the correctness of the solution but also reinforces our understanding of the equation and the problem-solving process. It's a valuable habit to develop, especially in mathematics, where accuracy is paramount. In this case, our verification confirms that y = -1 is indeed the correct solution when x = 2. So, with the solution verified, we can be confident in our answer.
Conclusion
In conclusion, we have successfully solved the equation 4x - 3y = 11 to find that y = -1 when x = 2. This process involved substitution, simplification, isolation of the variable, and finally, solving for y. We also verified our solution to ensure its accuracy. Solving linear equations is a fundamental skill in mathematics, and this example demonstrates the key steps involved. Understanding these steps allows us to tackle a wide range of algebraic problems. The ability to solve equations is not just an academic exercise; it's a practical skill that has applications in various fields. From science and engineering to finance and economics, solving equations is essential for modeling and analyzing real-world scenarios. By mastering these skills, we empower ourselves to make informed decisions and solve problems effectively. This article has provided a comprehensive guide to solving this specific equation, but the principles and techniques discussed can be applied to many other equations as well. So, continue practicing and refining your problem-solving skills to excel in mathematics and beyond. The journey of learning mathematics is a continuous one, and each problem solved is a step forward.