Introduction to Solving Systems of Linear Equations
In the realm of mathematics, solving systems of linear equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. A system of linear equations comprises two or more equations involving the same variables, and the goal is to find values for these variables that satisfy all equations simultaneously. This often represents finding the point(s) where the lines (in the case of two variables) or planes (in higher dimensions) intersect. In this article, we will delve into solving the specific system of equations:
3x + 3y = -9
x + 3y = -19
We will explore different methods to find the values of x and y that make both equations true. Mastering these techniques is essential for understanding more advanced mathematical concepts and real-world problem-solving scenarios. There are several approaches to tackle systems of linear equations, each with its own advantages and suitability depending on the specific system at hand. Common methods include substitution, elimination, and graphical methods. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved directly. The elimination method, also known as the addition method, focuses on manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This again leads to a single equation with one variable. The graphical method involves plotting the equations on a coordinate plane and finding the point(s) of intersection, which represent the solutions. Choosing the most efficient method often depends on the structure of the equations and personal preference. In this article, we will primarily focus on the elimination method, as it is particularly well-suited for this specific system of equations. Understanding the underlying principles of each method allows you to adapt and apply them to a wide range of problems. Furthermore, we will discuss how to verify our solution to ensure accuracy, a crucial step in any mathematical problem-solving process. By the end of this discussion, you will have a solid understanding of how to solve this system of equations and be equipped to tackle similar problems with confidence.
Method 1: Elimination Method
The elimination method, a cornerstone technique in solving systems of linear equations, shines when the coefficients of one variable in the equations are either the same or can be easily made the same. This method hinges on the principle of adding or subtracting multiples of equations to eliminate one variable, thereby reducing the problem to a single equation with a single unknown. Let's apply this method to our system:
3x + 3y = -9 (Equation 1)
x + 3y = -19 (Equation 2)
Notice that the coefficient of y is the same in both equations (3y). This makes the elimination method particularly appealing. Our strategy is to subtract Equation 2 from Equation 1, which will eliminate the y variable. This subtraction is performed term by term:
(3x + 3y) - (x + 3y) = -9 - (-19)
This simplifies to:
3x + 3y - x - 3y = -9 + 19
Combining like terms, we get:
2x = 10
Now, we can easily solve for x by dividing both sides by 2:
x = 10 / 2
x = 5
We have successfully found the value of x. Now that we know x, we can substitute it back into either Equation 1 or Equation 2 to solve for y. Let's choose Equation 2 for simplicity:
x + 3y = -19
Substitute x = 5:
5 + 3y = -19
Subtract 5 from both sides:
3y = -19 - 5
3y = -24
Divide both sides by 3:
y = -24 / 3
y = -8
Therefore, the solution to the system of equations using the elimination method is x = 5 and y = -8. This means that the point (5, -8) is the intersection of the two lines represented by the equations. It's crucial to verify this solution by substituting these values back into both original equations to ensure they hold true. This step confirms the accuracy of our calculations and provides confidence in our answer. The beauty of the elimination method lies in its systematic approach, making it a reliable technique for solving systems of equations, especially when coefficients are easily aligned for elimination.
Method 2: Substitution Method
The substitution method offers an alternative approach to solving systems of linear equations. This technique involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with one variable, which can be solved directly. Let's apply this method to our system:
3x + 3y = -9 (Equation 1)
x + 3y = -19 (Equation 2)
To begin, we need to choose one equation and solve for one of the variables. Equation 2 appears simpler for this purpose, so let's solve it for x:
x + 3y = -19
Subtract 3y from both sides:
x = -19 - 3y
Now we have an expression for x in terms of y. Next, we substitute this expression into Equation 1:
3x + 3y = -9
Substitute x = -19 - 3y:
3(-19 - 3y) + 3y = -9
Distribute the 3:
-57 - 9y + 3y = -9
Combine like terms:
-57 - 6y = -9
Add 57 to both sides:
-6y = -9 + 57
-6y = 48
Divide both sides by -6:
y = 48 / -6
y = -8
We have found the value of y. Now, substitute this value back into the expression we found for x:
x = -19 - 3y
Substitute y = -8:
x = -19 - 3(-8)
x = -19 + 24
x = 5
Thus, using the substitution method, we find the solution to be x = 5 and y = -8, which aligns with the solution we obtained using the elimination method. This reinforces the correctness of our answer. The substitution method is particularly useful when one equation can be easily solved for one variable in terms of the other. It provides a systematic way to reduce the system to a single equation, making it a valuable tool in solving linear systems. As with any method, it's essential to verify the solution by substituting the values back into the original equations to ensure accuracy.
Verification of the Solution
Verifying the solution is a crucial step in solving any mathematical problem, particularly when dealing with systems of equations. It ensures that the values we have found for the variables satisfy all the equations in the system simultaneously. This process acts as a safeguard against errors made during the solution process, whether they are arithmetic mistakes or conceptual misunderstandings. In our case, we found the solution to the system:
3x + 3y = -9 (Equation 1)
x + 3y = -19 (Equation 2)
to be x = 5 and y = -8. To verify this solution, we will substitute these values back into both Equation 1 and Equation 2 and check if the equations hold true.
Let's start with Equation 1:
3x + 3y = -9
Substitute x = 5 and y = -8:
3(5) + 3(-8) = -9
15 - 24 = -9
-9 = -9
The equation holds true. Now, let's verify with Equation 2:
x + 3y = -19
Substitute x = 5 and y = -8:
5 + 3(-8) = -19
5 - 24 = -19
-19 = -19
This equation also holds true. Since the values x = 5 and y = -8 satisfy both equations in the system, we can confidently conclude that this is the correct solution. This verification process not only confirms the correctness of our answer but also deepens our understanding of the relationship between the equations and their solution. It reinforces the concept that the solution to a system of equations represents the point of intersection of the lines (or planes in higher dimensions) represented by the equations. In practical applications, verifying the solution is especially important, as it ensures that the results obtained are reliable and can be used for further analysis or decision-making. By consistently incorporating this step into our problem-solving routine, we can minimize errors and build a strong foundation in mathematical reasoning.
Conclusion
In conclusion, we have successfully solved the system of linear equations:
3x + 3y = -9
x + 3y = -19
using two distinct methods: the elimination method and the substitution method. Both methods yielded the same solution, x = 5 and y = -8, which we subsequently verified by substituting these values back into the original equations. This comprehensive approach not only provided us with the solution but also reinforced our understanding of the underlying principles of solving systems of linear equations. The elimination method proved to be particularly efficient in this case due to the matching coefficients of the y variable, allowing for a straightforward elimination of one variable. The substitution method, on the other hand, offered a flexible alternative, showcasing its utility when one equation can be easily solved for one variable in terms of the other. The ability to solve the same problem using multiple methods is a testament to the versatility of mathematical techniques and enhances our problem-solving skills. Furthermore, the verification step is a critical component of the solution process, ensuring the accuracy and reliability of our results. It highlights the importance of rigorousness in mathematical reasoning and provides a safety net against potential errors. The skills and concepts discussed in this article are fundamental to a wide range of mathematical and scientific disciplines. Solving systems of equations is a core skill in algebra and calculus, and it has applications in fields such as physics, engineering, economics, and computer science. Mastering these techniques not only equips us to solve specific problems but also develops our analytical and problem-solving abilities, which are valuable assets in any field. By understanding the principles behind these methods and practicing their application, we can confidently tackle more complex problems and appreciate the power and elegance of mathematics. The journey of solving mathematical problems is not just about finding the right answer; it's about developing a logical and systematic approach, understanding the underlying concepts, and appreciating the beauty of mathematical reasoning.
