Verifying The Solution To A System Of Equations (-2,-1)
In the realm of mathematics, solving systems of equations is a fundamental skill, particularly in algebra. 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 for the variables that satisfy all equations simultaneously. This means that when these values are substituted into each equation, they make the equation true. In this article, we will delve into a specific system of equations and demonstrate how to verify its solution. The given system of equations is:
It is stated that the solution to this system is (-2, -1). To confirm this, we must substitute x = -2 and y = -1 into both equations and check if the equations hold true. This process of substitution is a cornerstone of solving algebraic problems, allowing us to test potential solutions systematically. Beyond verification, understanding how solutions are derived is crucial. Techniques such as substitution, elimination, and matrix methods are commonly used to find these solutions. Each method offers a unique approach, and the choice of method often depends on the specific characteristics of the equations themselves. For instance, the elimination method, which involves manipulating equations to cancel out variables, is particularly useful when coefficients are easily matched. Moreover, graphical methods offer a visual representation of solutions, showing where the lines or curves of the equations intersect. The ability to solve systems of equations is not just an academic exercise; it has practical applications in various fields, including engineering, economics, and computer science. From modeling physical systems to optimizing business processes, the principles of solving equations are fundamental. Therefore, mastering these techniques provides a solid foundation for more advanced mathematical studies and real-world problem-solving.
To verify the proposed solution (-2, -1) for the given system of equations, we will substitute x = -2 and y = -1 into each equation. This process involves replacing the variables x and y with their respective values and then performing the arithmetic operations to see if the equations hold true. The first equation is 2x - 3y = -1. Substituting x = -2 and y = -1, we get:
Since the result is -1, which matches the right-hand side of the equation, the solution (-2, -1) satisfies the first equation. Now, let’s substitute these values into the second equation, which is 11x - 9y = -13. Plugging in x = -2 and y = -1, we have:
Again, the result is -13, which is the same as the right-hand side of the equation. This confirms that the solution (-2, -1) also satisfies the second equation. Because the solution (-2, -1) satisfies both equations simultaneously, we can confidently conclude that it is indeed the correct solution to the system. This method of verification is essential in mathematics to ensure the accuracy of solutions. It provides a concrete way to check whether the calculated values for the variables are correct. Furthermore, understanding this verification process is crucial for solving more complex problems in algebra and beyond. The ability to accurately check solutions not only reinforces the understanding of the underlying mathematical principles but also builds confidence in one's problem-solving abilities. In addition to direct substitution, graphical methods and numerical techniques can also be used to verify solutions, especially in cases involving non-linear equations or larger systems. However, for linear systems, direct substitution remains one of the most straightforward and reliable methods.
The problem states that when the first equation is multiplied by -3, the sum of the two equations is equivalent to 5x = -10. To verify this, we first need to multiply the first equation by -3. The first equation is 2x - 3y = -1. Multiplying both sides by -3, we get:
Now, we will add this modified equation to the second equation, which is 11x - 9y = -13. Adding the two equations (-6x + 9y = 3) and (11x - 9y = -13), we have:
Combining like terms, we get:
This confirms that the sum of the two equations, after multiplying the first equation by -3, is indeed equivalent to 5x = -10. This process illustrates a key method in solving systems of equations, known as the elimination method. The goal is to manipulate the equations so that when they are added together, one of the variables is eliminated, making it easier to solve for the remaining variable. In this case, multiplying the first equation by -3 and then adding it to the second equation eliminated the y variable, leaving us with an equation in terms of x only. The elimination method is particularly useful when the coefficients of one of the variables in the two equations are multiples of each other, or when they have opposite signs. This allows for a straightforward cancellation of terms. Understanding and applying this method effectively is a valuable skill in algebra and beyond, as it provides a systematic approach to solving complex systems of equations. Furthermore, the ability to manipulate equations in this way highlights the importance of algebraic properties and the rules of arithmetic operations in mathematical problem-solving.
Having derived the equation 5x = -10, we can now solve for x. To do this, we need to isolate x by dividing both sides of the equation by 5. This is a fundamental algebraic operation based on the principle that performing the same operation on both sides of an equation maintains the equality. So, we divide both sides of 5x = -10 by 5:
This simplifies to:
Thus, we find that x = -2. This result aligns with the x-coordinate of the given solution (-2, -1). Solving for a single variable in a simplified equation is a crucial step in many algebraic problems. It demonstrates the power of inverse operations in isolating variables and finding their values. In this case, division is the inverse operation of multiplication, and by dividing both sides by the coefficient of x, we were able to determine its value. The ability to solve such simple equations is foundational for tackling more complex algebraic challenges. It reinforces the understanding of basic algebraic principles and provides a building block for more advanced techniques. Furthermore, this process highlights the importance of precision and attention to detail in mathematical calculations. A small error in arithmetic can lead to an incorrect solution, emphasizing the need for careful execution of each step. In the context of solving systems of equations, finding the value of one variable is often the first step towards finding the complete solution. Once one variable is known, it can be substituted into one of the original equations to solve for the remaining variable, as we saw in the verification process earlier.
We have shown that the solution to the system of equations is indeed (-2, -1). We verified this by substituting x = -2 and y = -1 into both equations and confirming that they hold true. Additionally, we demonstrated that when the first equation is multiplied by -3, the sum of the resulting equation and the second equation is equivalent to 5x = -10. Solving this equation, we found x = -2, which matches the x-coordinate of the given solution. This comprehensive process not only verifies the solution but also illustrates key methods used in solving systems of equations, such as substitution and elimination. The fact that we were able to derive 5x = -10 through algebraic manipulation and then solve for x reinforces the consistency and interconnectedness of mathematical principles. It also highlights the importance of understanding different approaches to problem-solving, as the elimination method provided a straightforward way to find the value of x. In mathematical problem-solving, there are often multiple paths to the solution, and choosing the most efficient method can save time and effort. The ability to recognize and apply different techniques is a sign of mathematical fluency and expertise. Moreover, the verification process underscores the need for accuracy and thoroughness in mathematical work. Checking the solution by substituting it back into the original equations is a crucial step that helps prevent errors and builds confidence in the final result. This practice is not only applicable to solving systems of equations but also to various other mathematical problems. In conclusion, the given solution (-2, -1) is confirmed to be correct, and the steps taken to verify and derive this solution demonstrate fundamental algebraic principles and problem-solving strategies.
In summary, we have thoroughly examined the system of equations and verified that the solution (-2, -1) is accurate. This was achieved through direct substitution into the original equations and by demonstrating the algebraic manipulation that leads to the equation 5x = -10. Solving 5x = -10, we confirmed that x = -2, which aligns with the provided solution. The process of solving systems of equations is a cornerstone of algebra and has far-reaching applications in various fields. Understanding techniques such as substitution and elimination is crucial for students and professionals alike. These methods provide systematic approaches to problem-solving and are applicable to a wide range of mathematical challenges. Moreover, the ability to verify solutions is a vital skill that ensures accuracy and builds confidence in one's mathematical abilities. The steps taken in this analysis highlight the interconnectedness of algebraic principles and the importance of logical reasoning in mathematics. From manipulating equations to isolating variables, each step relies on fundamental mathematical rules and properties. By mastering these concepts, individuals can approach complex problems with clarity and precision. Furthermore, this exercise underscores the significance of practice in mathematics. The more one engages with problem-solving, the more proficient they become in recognizing patterns, applying techniques, and verifying solutions. In conclusion, the solution (-2, -1) to the given system of equations has been rigorously verified, and the methods employed demonstrate essential algebraic skills and problem-solving strategies.
