Solution Of The System Of Linear Equations
In this article, we will delve into the process of solving a system of linear equations. Linear equations are fundamental in mathematics and have wide applications in various fields such as physics, engineering, economics, and computer science. Understanding how to solve these systems is crucial for anyone pursuing studies or careers in these areas. We will explore a specific system of two linear equations with two variables and determine the correct solution step by step. The system we are tackling is:
-3x + 4y = -18
2x - y = 7
The task is to find the values of x and y that satisfy both equations simultaneously. We will examine different methods to solve this system, including substitution, elimination, and graphical methods, to ensure a comprehensive understanding. Let's dive in and find the correct solution among the given options:
Understanding Systems of Linear Equations
Before we jump into solving the given system, let's first understand what systems of linear equations are and why they are important. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variables are only raised to the power of one. A system of linear equations is a collection of two or more linear equations involving the same set of variables. The solution to a system of linear equations is the set of values for the variables that make all the equations true simultaneously. This solution represents the point(s) where the lines represented by the equations intersect on a graph. Systems of linear equations can have one solution, no solution, or infinitely many solutions, depending on the relationships between the lines.
Solving linear equations is a fundamental skill in algebra, and it forms the basis for solving more complex problems in various fields. In mathematics, these systems arise in numerous contexts, from basic algebra to advanced calculus and linear algebra. In physics, they can be used to model various phenomena, such as electrical circuits, motion, and equilibrium. In economics, they are used in supply and demand models, cost analysis, and optimization problems. In computer science, they are used in areas like computer graphics, network analysis, and machine learning. Therefore, mastering the techniques for solving systems of linear equations is essential for anyone working in these fields. The most common methods for solving linear systems are substitution, elimination, and graphical methods. Each method has its advantages and disadvantages, and the choice of method often depends on the specific system of equations. For instance, the substitution method is effective when one equation can easily be solved for one variable in terms of the other. The elimination method is particularly useful when the coefficients of one variable in the equations are opposites or can be made opposites through multiplication. Graphical methods provide a visual representation of the solution, which can be especially helpful for understanding the nature of the solutions. In our specific case, we will explore the substitution and elimination methods to determine the solution.
Methods to Solve Systems of Linear Equations
To solve the system of linear equations:
-3x + 4y = -18
2x - y = 7
We can use several methods, including substitution, elimination, and graphing. Here, we will focus on the substitution and elimination methods, as they are the most direct for this particular system.
1. Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation in one variable, which can then be easily solved. Let's apply this method to our system.
First, we choose one of the equations to solve for one variable. The second equation, 2x - y = 7, looks simpler to solve for y. We can rewrite it as:
y = 2x - 7
Now, we substitute this expression for y into the first equation:
-3x + 4(2x - 7) = -18
Expanding and simplifying the equation, we get:
-3x + 8x - 28 = -18
5x = 10
x = 2
Now that we have the value of x, we can substitute it back into the expression for y:
y = 2(2) - 7
y = 4 - 7
y = -3
So, the solution using the substitution method is x = 2 and y = -3. This gives us the ordered pair (2, -3). The substitution method is particularly effective when one of the equations can be easily solved for one variable. It involves substituting the expression for one variable into the other equation, which reduces the problem to solving a single equation in one variable. After finding the value of one variable, we substitute it back into the expression to find the value of the other variable. The steps are straightforward and methodical, making it a reliable approach for solving linear systems. In cases where equations are not easily solved for a variable, the elimination method might be more efficient. However, the substitution method is a powerful tool in the arsenal of techniques for solving linear equations. The key to success with this method is careful algebraic manipulation and attention to detail to avoid errors in substitution and simplification.
2. Elimination Method
The elimination method, also known as the addition method, involves manipulating the equations so that when they are added together, one of the variables is eliminated. This also reduces the system to a single equation in one variable. Let's apply this method to our system.
We have the system:
-3x + 4y = -18
2x - y = 7
To eliminate y, we can multiply the second equation by 4:
4(2x - y) = 4(7)
8x - 4y = 28
Now, we add the modified second equation to the first equation:
(-3x + 4y) + (8x - 4y) = -18 + 28
5x = 10
x = 2
Now that we have the value of x, we can substitute it back into either of the original equations to find y. Let's use the second equation:
2(2) - y = 7
4 - y = 7
y = -3
So, the solution using the elimination method is x = 2 and y = -3. This also gives us the ordered pair (2, -3). The elimination method is a powerful technique, particularly effective when the coefficients of one variable in the two equations are the same or easily made the same by multiplication. The core idea is to manipulate the equations such that adding them cancels out one of the variables, simplifying the system to a single equation in one variable. This approach often requires multiplying one or both equations by suitable constants to match the coefficients of the variable to be eliminated. Once the equations are in the appropriate form, adding them results in a new equation with only one variable, which can then be solved directly. After finding the value of one variable, it can be substituted back into any of the original or modified equations to find the value of the other variable. The elimination method is especially advantageous when the equations have coefficients that make substitution cumbersome, as it provides a systematic way to eliminate variables and solve the system.
Comparing the Methods
Both the substitution and elimination methods are effective ways to solve systems of linear equations, but the choice between them often depends on the specific structure of the equations. The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. This makes it straightforward to substitute the expression into the other equation, reducing the problem to a single equation in one variable. On the other hand, the elimination method shines when the coefficients of one variable in the two equations are the same or can be easily made the same by multiplication. This allows for the elimination of that variable when the equations are added or subtracted, again simplifying the system to a single equation in one variable.
In the case of the given system:
-3x + 4y = -18
2x - y = 7
The substitution method was slightly more straightforward because the second equation could be easily solved for y. However, the elimination method was also quite efficient once we multiplied the second equation by 4 to eliminate y. Both methods led us to the same solution, demonstrating their consistency and reliability.
Choosing between the two often comes down to personal preference and the specific details of the problem. Some people find substitution more intuitive, while others prefer the systematic approach of elimination. Regardless of the method chosen, the key is to apply it carefully and accurately, paying close attention to algebraic manipulations and signs. Both methods provide valuable tools for solving systems of linear equations, and proficiency in both is beneficial for tackling a wide range of problems. Ultimately, the goal is to find the values of the variables that satisfy all equations in the system simultaneously, and both substitution and elimination are effective ways to achieve this.
Analyzing the Options
We have found the solution to the system of equations to be x = 2 and y = -3. Now, let's compare this solution with the given options:
A.
(-2, -3)
B.
(-2, 3)
C.
(2, -3)
D.
(2, 3)
Our solution (2, -3) matches option C. Therefore, the correct solution is:
C.
(2, -3)
The process of analyzing the options is a critical step in problem-solving, especially in mathematics. After arriving at a solution, it's essential to verify that the solution aligns with one of the given options and to ensure that no mistakes were made during the solving process. This step acts as a final check, reducing the likelihood of selecting an incorrect answer due to a simple error in calculation or transcription. In this case, we meticulously solved the system of linear equations using both the substitution and elimination methods, arriving at the solution x = 2 and y = -3. We then compared this solution with the provided options and found an exact match with option C, (2, -3). This alignment not only confirms the accuracy of our solution but also reinforces our confidence in the chosen method and the steps taken.
In situations where the calculated solution doesn't directly match any of the given options, it's prudent to revisit the solving process and check for any potential errors. It's also possible that the solution might need to be manipulated or simplified to match one of the options. The skill of option analysis is particularly valuable in multiple-choice questions, where the correct answer is already present among the choices. By carefully comparing the derived solution with the options, students can improve their accuracy and performance in mathematical problem-solving.
Conclusion
In conclusion, the solution to the system of linear equations:
-3x + 4y = -18
2x - y = 7
is (2, -3). We arrived at this solution using both the substitution and elimination methods, demonstrating the consistency and reliability of these techniques. Understanding and applying these methods is crucial for solving systems of linear equations, which have broad applications in various fields. This comprehensive exploration should provide a solid understanding of how to approach and solve such problems effectively.
The ability to solve systems of linear equations is a cornerstone of mathematical proficiency, extending its influence across numerous disciplines. In this article, we not only identified the solution to a specific system but also emphasized the importance of understanding the underlying principles and methods involved. The substitution and elimination methods, both powerful tools in their own right, offer different approaches to tackling these problems, and familiarity with both enhances problem-solving flexibility. The process of verifying the solution against the given options underscores the importance of precision and attention to detail in mathematics.
Moreover, the broader applications of linear systems in fields like physics, economics, and computer science highlight the practical relevance of this skill. Whether it's modeling physical phenomena, analyzing economic trends, or developing algorithms, the ability to solve systems of linear equations is an invaluable asset. This article has aimed not only to provide a solution to a particular problem but also to foster a deeper appreciation for the significance of linear algebra in the broader context of scientific and mathematical endeavors. By mastering these fundamental concepts, students and professionals alike can unlock a wide array of problem-solving capabilities and contribute meaningfully to their respective fields.