Solving System Of Equations 6y = -24 And X + 4y = 1
Introduction
In mathematics, solving a system of equations is a fundamental skill. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values that, when substituted for the variables, make all the equations true. There are several methods for solving systems of equations, including substitution, elimination, and graphing. This article will delve into solving the specific system of equations provided, showcasing a step-by-step approach and highlighting key concepts. We will address the nuances of dependent systems and how to express their solution sets. This exploration aims to provide a comprehensive understanding for students and enthusiasts alike, ensuring they can confidently tackle similar problems.
Understanding Systems of Equations
Before diving into the solution, let's first grasp the concept of a system of equations. A system of equations comprises two or more equations that share variables. The solution to such a system is the set of values for the variables that satisfy all equations simultaneously. Systems of equations can represent various real-world scenarios, from calculating the cost of items to determining the intersection points of geometric shapes. The most common types of systems involve linear equations, which, when graphed, produce straight lines. The solution to a system of two linear equations, for instance, corresponds to the point where the two lines intersect. However, systems can also involve non-linear equations, leading to more complex solutions. The nature of the solution set (unique, infinite, or none) depends on the relationship between the equations. For instance, parallel lines will never intersect, resulting in no solution, while overlapping lines indicate infinitely many solutions, a characteristic of dependent systems. Mastering the art of solving systems of equations is crucial not only in mathematics but also in fields like physics, engineering, and economics, where modeling and problem-solving often involve multiple variables and constraints.
The Given System of Equations
In this article, we will focus on solving the following system of equations:
6y = -24
x + 4y = 1
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations simultaneously. The first equation, 6y = -24, is a straightforward equation in one variable, making it an ideal starting point for solving the system. The second equation, x + 4y = 1, involves both variables and will require substitution or elimination techniques after we've determined the value of y. The structure of this system allows for a relatively simple solution process, as one variable can be easily isolated in the first equation. However, it's essential to consider the possibility of the system being dependent, which would mean that the equations represent the same line or parallel lines, leading to either infinitely many solutions or no solution, respectively. Therefore, after finding the initial solution, we will verify its validity by substituting the values back into both equations. This step ensures that we have indeed found the correct solution and helps us identify any potential errors in our calculations.
Step-by-Step Solution
Step 1: Solve the First Equation for y
The first equation is 6y = -24. To isolate y, we divide both sides of the equation by 6:
6y / 6 = -24 / 6
y = -4
This step is crucial as it immediately gives us the value of y, which we can then use in the subsequent steps to find the value of x. The simplicity of this equation allows for a direct solution, avoiding the need for more complex algebraic manipulations. By dividing both sides by the coefficient of y, we maintain the equality and successfully isolate the variable. This process demonstrates a fundamental principle in solving equations: performing the same operation on both sides ensures that the equation remains balanced. The result, y = -4, is a key piece of information that simplifies the second equation, allowing us to solve for x with ease. Understanding and executing this step correctly is essential for accurately solving the entire system of equations.
Step 2: Substitute the Value of y into the Second Equation
Now that we have found y = -4, we can substitute this value into the second equation, x + 4y = 1:
x + 4(-4) = 1
x - 16 = 1
Substituting the value of y into the second equation is a critical step in solving the system. This process effectively reduces the second equation to a single variable equation, making it solvable for x. By replacing y with its numerical value, we transform the equation from involving two variables to one, which is a standard technique in solving systems of equations. The resulting equation, x - 16 = 1, is now a simple linear equation that can be easily solved for x. This step highlights the power of substitution as a method for simplifying complex equations. The accuracy of this substitution is paramount, as any error here will propagate through the rest of the solution. Therefore, careful attention to detail is essential when performing this step. This substitution technique is a cornerstone of algebra and is widely used in various mathematical contexts beyond solving systems of equations.
Step 3: Solve for x
To solve for x in the equation x - 16 = 1, we add 16 to both sides:
x - 16 + 16 = 1 + 16
x = 17
Solving for x involves isolating the variable on one side of the equation. In this case, adding 16 to both sides cancels out the -16 term on the left, leaving x by itself. This process demonstrates the fundamental principle of maintaining equality by performing the same operation on both sides. The resulting value, x = 17, is the solution for the variable x in the system of equations. This step is straightforward but crucial, as it completes the process of finding the values for both variables. The accuracy of this step is vital, as the value of x will be part of the final solution set. This simple algebraic manipulation is a core skill in mathematics and is frequently used in solving various types of equations. With both x and y now determined, we can express the solution to the system of equations.
Step 4: Express the Solution Set
The solution to the system of equations is x = 17 and y = -4. We can express this as an ordered pair (x, y):
(17, -4)
This ordered pair represents the point where the two lines represented by the equations intersect on a graph. Expressing the solution as an ordered pair is a standard convention in mathematics, providing a clear and concise representation of the solution set. The order of the values is significant, with the x-coordinate listed first and the y-coordinate second. This notation is universally understood and facilitates communication of the solution in mathematical contexts. The ordered pair (17, -4) indicates that when x is 17 and y is -4, both equations in the system are satisfied. This solution can be visually verified by graphing the two equations and observing that they indeed intersect at the point (17, -4). The clarity and precision of expressing the solution as an ordered pair underscore the importance of standardized mathematical notation in ensuring unambiguous communication.
Checking the Solution
To ensure our solution is correct, we substitute x = 17 and y = -4 back into both original equations:
Equation 1: 6y = -24
6(-4) = -24
-24 = -24 (True)
Equation 2: x + 4y = 1
17 + 4(-4) = 1
17 - 16 = 1
1 = 1 (True)
Both equations hold true, so our solution (17, -4) is correct. This verification step is crucial in the problem-solving process. By substituting the values back into the original equations, we confirm that the solution satisfies all conditions of the system. This process helps identify any errors made during the solution process, such as incorrect algebraic manipulations or arithmetic mistakes. The satisfaction of both equations provides a high degree of confidence in the accuracy of the solution. This checking procedure is a fundamental aspect of mathematical rigor and is applicable to various problem-solving scenarios. The confirmation that the solution is correct reinforces the understanding of the concepts and techniques used in solving systems of equations. This step also highlights the importance of precision and attention to detail in mathematical calculations.
Conclusion
We have successfully solved the system of equations and found the solution to be (17, -4). This solution satisfies both equations in the system. The step-by-step approach outlined in this article provides a clear and concise method for solving similar systems of equations. The process involved isolating one variable, substituting its value into another equation, and then solving for the remaining variable. The final step of checking the solution ensures the accuracy and validity of the result. Understanding and applying these techniques are crucial for success in algebra and related mathematical disciplines. The ability to solve systems of equations is not only a fundamental mathematical skill but also a valuable tool in various fields, including science, engineering, and economics. By mastering these techniques, students and practitioners can confidently tackle a wide range of problems involving multiple variables and constraints. The solution process also underscores the importance of precision, attention to detail, and logical reasoning in mathematical problem-solving. The satisfaction of both equations confirms the correctness of the solution and reinforces the understanding of the underlying concepts.
Understanding Dependent Systems
Although the system we solved had a unique solution, it's important to understand the concept of dependent systems. A dependent system of equations is a system where the equations are essentially multiples of each other, resulting in an infinite number of solutions. This occurs when the equations represent the same line. When solving a dependent system, you will typically find that one equation reduces to an identity (e.g., 0 = 0), indicating that there are infinitely many solutions. To express the solution set for a dependent system, we typically solve one equation for one variable in terms of the other and then write the solution set as a set of ordered pairs. For example, if we had a system where the equations simplified to y = 2x + 1, the solution set would be expressed as {(x, y) | y = 2x + 1}, meaning any ordered pair that satisfies the equation y = 2x + 1 is a solution. Recognizing dependent systems is a crucial skill in solving systems of equations, as it requires a different approach for expressing the solution set compared to systems with unique solutions. The identification of dependent systems often involves simplifying the equations and observing whether they are scalar multiples of each other. This understanding is essential for a comprehensive grasp of linear algebra and its applications.
Solving System of Equations: Practice Problems
To solidify your understanding, consider trying to solve the following system of equations:
2x + y = 7
x - y = 2
This system can be solved using either substitution or elimination methods. Working through this practice problem will help reinforce the concepts and techniques discussed in this article. Remember to check your solution by substituting the values back into the original equations. Additionally, consider exploring other systems of equations with varying complexities to further develop your problem-solving skills. These practice problems should include cases with unique solutions, no solutions (inconsistent systems), and infinitely many solutions (dependent systems). The ability to recognize and solve different types of systems is a key skill in mathematics. By engaging in consistent practice, you can build confidence and proficiency in solving systems of equations. Furthermore, understanding the graphical interpretation of these systems, such as the intersection of lines, can provide a visual aid in comprehending the nature of the solutions. Practice is essential for mastering this fundamental mathematical concept.
Conclusion
Solving systems of equations is a fundamental skill in mathematics with applications in various fields. By following a step-by-step approach and understanding the concepts, you can confidently solve a wide range of systems. Remember to always check your solutions to ensure accuracy and to be aware of the possibility of dependent systems. With practice and persistence, you can master this essential mathematical skill. The ability to solve systems of equations is not only crucial for academic success but also for real-world problem-solving. The techniques discussed in this article provide a solid foundation for further exploration of linear algebra and related topics. The consistent application of these methods will enhance your mathematical reasoning and problem-solving abilities. Furthermore, the understanding of systems of equations opens doors to more advanced mathematical concepts and applications. This skill is an invaluable asset in various disciplines, highlighting its importance in a well-rounded education.
