Solving Systems Of Equations A Detailed Guide
Introduction
In mathematics, solving a system of equations is a fundamental skill with applications across various fields, including engineering, economics, and computer science. A system of equations is a set of two or more equations containing the same variables. The solution to the system is the set of values for the variables that satisfy all equations simultaneously. This article delves into the process of solving a specific system of linear equations, providing a step-by-step guide to finding the values of the variables that make the equations true. We will explore the methods used to manipulate the equations, isolate variables, and ultimately arrive at the solution. Understanding these techniques is crucial for anyone working with mathematical models and real-world problems that can be represented by such systems. The focus will be on clarity and precision, ensuring that readers can follow each step and apply the same principles to other similar problems. This detailed explanation aims to enhance comprehension and mastery of this essential mathematical skill.
Understanding the Equations
The given system of equations consists of two linear equations with two variables, x and y. The equations are:
- -9y + 4x - 11 = 0
- -3y + 10x + 31 = 0
These equations represent straight lines on a coordinate plane. The solution to the system is the point (x, y) where these lines intersect. To find this point, we need to solve for x and y such that both equations are satisfied. There are several methods to solve such systems, including substitution, elimination, and matrix methods. In this article, we will primarily focus on the elimination method, which involves manipulating the equations to eliminate one variable, allowing us to solve for the other. Then, we will substitute the value obtained back into one of the original equations to find the value of the eliminated variable. This method is particularly effective when the coefficients of one of the variables are multiples of each other, making it easier to eliminate that variable.
Before diving into the solution, it's essential to understand the structure of linear equations and how they behave. A linear equation in two variables can be written in the general form Ax + By + C = 0, where A, B, and C are constants. The coefficients A and B determine the slope and orientation of the line, while C affects its position on the coordinate plane. By rearranging the equations and performing algebraic operations, we can systematically isolate the variables and find their values. The process involves careful attention to detail and a solid understanding of algebraic principles. This foundation is crucial for tackling more complex systems of equations and mathematical problems.
Method 1: Elimination Method
The elimination method is a powerful technique for solving systems of equations. The main idea behind this method is to manipulate the equations in such a way that when they are added or subtracted, one of the variables is eliminated. This leaves us with a single equation in one variable, which can be easily solved. Once we find the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable.
To apply the elimination method to our system:
- -9y + 4x - 11 = 0
- -3y + 10x + 31 = 0
We first need to find a common multiple for either the coefficients of x or the coefficients of y. In this case, it's easier to eliminate y because the coefficients -9 and -3 have a common multiple of 9. We can multiply the second equation by -3 to make the coefficient of y equal to 9:
-3 * (-3y + 10x + 31) = -3 * 0
This gives us:
9y - 30x - 93 = 0
Now we have the following system:
- -9y + 4x - 11 = 0
- 9y - 30x - 93 = 0
Next, we add the two equations together. This will eliminate the y variable:
(-9y + 4x - 11) + (9y - 30x - 93) = 0 + 0
Simplifying, we get:
-26x - 104 = 0
Now, we can solve for x:
-26x = 104
x = 104 / -26
x = -4
Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the first equation:
-9y + 4*(-4) - 11 = 0
-9y - 16 - 11 = 0
-9y - 27 = 0
-9y = 27
y = 27 / -9
y = -3
Thus, the solution to the system of equations is x = -4 and y = -3. This point represents the intersection of the two lines on the coordinate plane. The elimination method provides a systematic way to solve systems of equations by strategically manipulating the equations to eliminate variables, making it a valuable tool in mathematical problem-solving.
Method 2: Substitution Method
The substitution method is another effective technique for solving systems of equations. Unlike the elimination method, which aims to eliminate one variable by manipulating the equations, the substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process results in a single equation with one variable, which can be easily solved. Once the value of this variable is found, it can be substituted back into one of the original equations to determine the value of the other variable.
To apply the substitution method to our system:
- -9y + 4x - 11 = 0
- -3y + 10x + 31 = 0
We first need to solve one of the equations for one variable. Let's solve the first equation for x:
-9y + 4x - 11 = 0
4x = 9y + 11
x = (9y + 11) / 4
Now, we substitute this expression for x into the second equation:
-3y + 10*((9y + 11) / 4) + 31 = 0
To simplify, we multiply both sides of the equation by 4 to eliminate the fraction:
4*(-3y + 10*((9y + 11) / 4) + 31) = 4 * 0
-12y + 10*(9y + 11) + 124 = 0
-12y + 90y + 110 + 124 = 0
78y + 234 = 0
Now, we can solve for y:
78y = -234
y = -234 / 78
y = -3
Now that we have the value of y, we substitute it back into the expression for x:
x = (9*(-3) + 11) / 4
x = (-27 + 11) / 4
x = -16 / 4
x = -4
Thus, the solution to the system of equations is x = -4 and y = -3, which matches the solution obtained using the elimination method. The substitution method is particularly useful when one of the equations can be easily solved for one variable, making it a flexible and valuable tool for solving systems of equations.
Verification of the Solution
After solving a system of equations, it is crucial to verify the solution to ensure its accuracy. This involves substituting the values obtained for the variables back into the original equations to check if they satisfy both equations simultaneously. This step helps to catch any errors made during the solution process, such as arithmetic mistakes or incorrect algebraic manipulations. Verifying the solution provides confidence in the result and ensures that it is indeed the correct answer.
To verify our solution x = -4 and y = -3, we substitute these values into the original equations:
- -9y + 4x - 11 = 0
- -3y + 10x + 31 = 0
Substituting into the first equation:
-9*(-3) + 4*(-4) - 11 = 0
27 - 16 - 11 = 0
0 = 0
The first equation is satisfied.
Substituting into the second equation:
-3*(-3) + 10*(-4) + 31 = 0
9 - 40 + 31 = 0
0 = 0
The second equation is also satisfied.
Since both equations are satisfied by the values x = -4 and y = -3, we can confidently conclude that this is the correct solution to the system of equations. This verification step is a critical part of the problem-solving process and should not be overlooked. It reinforces the understanding of the solution and ensures that it is mathematically sound. The verification process highlights the importance of precision and attention to detail in solving mathematical problems, as even a small error can lead to an incorrect result. By taking the time to verify the solution, we can ensure that our answer is accurate and reliable.
Conclusion
In conclusion, solving systems of equations is a fundamental mathematical skill with wide-ranging applications. In this article, we explored two common methods for solving a system of linear equations: the elimination method and the substitution method. Both methods are effective and provide a systematic approach to finding the values of the variables that satisfy the equations. The elimination method involves manipulating the equations to eliminate one variable, while the substitution method involves solving one equation for one variable and substituting that expression into the other equation.
We applied both methods to the system of equations:
- -9y + 4x - 11 = 0
- -3y + 10x + 31 = 0
and found the solution to be x = -4 and y = -3 using both methods. This consistency in results reinforces the accuracy of our solution and demonstrates the versatility of these methods.
Additionally, we emphasized the importance of verifying the solution by substituting the values back into the original equations. This step ensures that the solution is correct and helps to catch any errors made during the problem-solving process. The verification process is a crucial part of mathematical problem-solving and should always be included as a final step.
Understanding and mastering these techniques for solving systems of equations is essential for students and professionals in various fields. These skills are not only applicable to mathematical problems but also to real-world scenarios that can be modeled using linear equations. The ability to solve systems of equations efficiently and accurately is a valuable asset in problem-solving and decision-making.
By following the step-by-step guides provided in this article and practicing with various examples, readers can enhance their understanding and proficiency in solving systems of equations. The knowledge and skills gained from this article will serve as a solid foundation for tackling more complex mathematical problems in the future. The key takeaways are the importance of systematic approaches, the versatility of different methods, and the necessity of verification to ensure accuracy. With practice and attention to detail, anyone can master the art of solving systems of equations.
