Solving System Of Equations: A Step-by-Step Guide
Introduction
In the realm of mathematics, solving a system of equations is a fundamental skill. It involves finding the values of the variables that satisfy all the equations in the system simultaneously. This guide focuses on providing a detailed, step-by-step explanation of how to solve the given system of equations. Mastering this process will enhance your problem-solving abilities and open doors to more advanced mathematical concepts. To effectively solve the system of equations, we will explore methods such as substitution and elimination. Understanding these techniques is crucial for tackling various mathematical problems, making this guide an invaluable resource for students and anyone looking to sharpen their math skills.
Understanding Systems of Equations
Before diving into the solution, it's essential to understand what a system of equations is. 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 make all the equations true. Graphically, the solution represents the point(s) where the lines (or curves, in more complex systems) intersect. Systems of equations are applicable in various real-world scenarios, from calculating financial investments to determining the trajectory of projectiles. The beauty of mathematics lies in its ability to model and solve real-life problems, and understanding systems of equations is a key part of this. There are several methods to solve a system of equations, including graphing, substitution, and elimination. Each method has its advantages and is suitable for different types of systems. The choice of method often depends on the structure of the equations and personal preference. In this guide, we will primarily focus on algebraic methods, as they provide a precise and efficient way to find solutions. A solid grasp of these methods empowers you to tackle more complex mathematical challenges with confidence.
The Given System of Equations
Let's consider the system of equations we aim to solve:
6x + 2y = 12
x - y = 2/3
This system comprises two linear equations, each containing two variables, x and y. Linear equations, when graphed, represent straight lines. The solution to this system will be the point of intersection of these two lines, representing the values of x and y that satisfy both equations. To effectively solve this system, we can use several algebraic methods, such as substitution or elimination. The first equation, 6x + 2y = 12, can be simplified by dividing all terms by 2, which results in 3x + y = 6. This simplification makes the equation easier to work with in subsequent steps. The second equation, x - y = 2/3, represents another straight line. The challenge lies in finding the specific values of x and y that make both equations true simultaneously. This is where the power of algebraic techniques comes into play. By manipulating the equations and isolating variables, we can systematically arrive at the solution. Understanding the nature of these equations and their graphical representation provides a strong foundation for tackling the problem at hand. The steps involved in solving this system will be outlined in the following sections, offering a clear and concise approach to finding the values of x and y.
Method 1: Solving by Substitution
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved. Let's apply this method to our system:
6x + 2y = 12
x - y = 2/3
First, we'll solve the second equation for x:
x = y + 2/3
Now, substitute this expression for x into the first equation:
6(y + 2/3) + 2y = 12
Distribute the 6:
6y + 4 + 2y = 12
Combine like terms:
8y + 4 = 12
Subtract 4 from both sides:
8y = 8
Divide by 8:
y = 1
Now that we have the value of y, we can substitute it back into the expression for x:
x = 1 + 2/3
x = 5/3
Therefore, the solution to the system of equations using the substitution method is x = 5/3 and y = 1. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. The key is to carefully substitute the expression into the other equation and simplify. By following this systematic approach, we can efficiently find the solution to the system. The substitution method is a powerful tool in solving systems of equations, and understanding it can significantly enhance your problem-solving abilities.
Method 2: Solving by Elimination
Another effective method for solving systems of equations is the elimination method. This technique involves manipulating the equations so that the coefficients of one variable are opposites, allowing you to eliminate that variable by adding the equations together. Let's apply the elimination method to our system:
6x + 2y = 12
x - y = 2/3
First, we need to make the coefficients of either x or y opposites. Let's choose to eliminate y. To do this, multiply the second equation by 2:
2(x - y) = 2(2/3)
2x - 2y = 4/3
Now, we have the following system:
6x + 2y = 12
2x - 2y = 4/3
Add the two equations together:
(6x + 2y) + (2x - 2y) = 12 + 4/3
Simplify:
8x = 40/3
Divide by 8:
x = (40/3) / 8
x = 5/3
Now that we have the value of x, we can substitute it back into one of the original equations to find y. Let's use the second equation:
(5/3) - y = 2/3
Solve for y:
y = (5/3) - (2/3)
y = 1
Therefore, the solution to the system of equations using the elimination method is x = 5/3 and y = 1. This method is particularly effective when the coefficients of one variable are easily made opposites. The elimination method offers a systematic way to solve systems of equations, and understanding this technique can greatly improve your mathematical skills. By mastering both the substitution and elimination methods, you'll be well-equipped to tackle a wide range of problems.
Solution Verification
After solving a system of equations, it's crucial to verify the solution to ensure its accuracy. This step involves substituting the values of x and y back into the original equations to confirm that they satisfy both. Our solution is x = 5/3 and y = 1. Let's substitute these values into the original equations:
Equation 1: 6x + 2y = 12
6(5/3) + 2(1) = 12
10 + 2 = 12
12 = 12 (True)
Equation 2: x - y = 2/3
(5/3) - 1 = 2/3
(5/3) - (3/3) = 2/3
2/3 = 2/3 (True)
Since the values of x and y satisfy both equations, our solution is correct. Verifying the solution is an essential step in the problem-solving process. It not only ensures the accuracy of your answer but also reinforces your understanding of the equations and the solution process. This practice is highly recommended for all mathematical problems, as it helps prevent errors and builds confidence in your results. By taking the time to verify your solution, you can be certain that you have found the correct answer and have a solid grasp of the concepts involved. This meticulous approach is a hallmark of strong mathematical skills and problem-solving abilities.
Conclusion
In conclusion, we have successfully solved the system of equations:
6x + 2y = 12
x - y = 2/3
using both the substitution and elimination methods. The solution, x = 5/3 and y = 1, satisfies both equations, as verified. Mastering the techniques of solving systems of equations is a fundamental skill in mathematics, with applications ranging from basic algebra to advanced calculus and real-world problem-solving. The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This method is particularly useful when one equation can be easily solved for a variable. The elimination method, on the other hand, involves manipulating the equations to eliminate one variable, making it easier to solve for the other. This method is effective when the coefficients of one variable are opposites or can be easily made opposites. By understanding and practicing both methods, you can develop a versatile toolkit for tackling various systems of equations. Remember, verification of the solution is a critical step to ensure accuracy. With a solid grasp of these concepts, you'll be well-prepared to handle more complex mathematical challenges. For further learning and practice, explore resources like Khan Academy's Systems of Equations section. This will provide you with additional examples and exercises to reinforce your understanding. Happy problem-solving!
