Solving For X In The System Of Equations 6x + 3y = 9 And 3x + 3y = 3

In the realm of mathematics, solving systems of equations is a fundamental skill. These systems, comprising two or more equations with shared variables, challenge us to find values that satisfy all equations simultaneously. One common method for tackling these problems is through elimination, where we manipulate the equations to cancel out one variable, allowing us to solve for the other. This article delves into the step-by-step process of solving a system of equations using the elimination method, with a focus on the example: 6x + 3y = 9 and 3x + 3y = 3. We will explore the underlying principles, the practical application of the method, and the importance of verifying the solution. By understanding these concepts, you'll gain a solid foundation for tackling a wide range of mathematical problems.

Understanding Systems of Equations

Before diving into the solution, let's define what a system of equations is. A system of equations is a set of two or more equations that share one or more variables. The goal is to find values for the variables that make all equations true. These values, when substituted into the original equations, should satisfy each equation simultaneously. Systems of equations can be found in various branches of mathematics and have practical applications in fields such as physics, engineering, economics, and computer science. The number of solutions a system has depends on the equations themselves. Some systems have one unique solution, others have infinitely many solutions, and some have no solutions at all. The process of solving a system involves using algebraic manipulations to isolate the variables and find their values. This may involve substitution, elimination, graphing, or matrix methods, depending on the complexity of the system.

Linear Equations and Their Representations

Linear equations, the building blocks of many systems, have a specific form: ax + by = c, where 'a,' 'b,' and 'c' are constants, and 'x' and 'y' are variables. These equations represent straight lines when graphed on a coordinate plane. The solutions to a system of linear equations correspond to the points where the lines intersect. If the lines intersect at one point, the system has a unique solution. If the lines are parallel, they never intersect, and the system has no solution. If the lines coincide (are the same line), there are infinitely many solutions. Understanding the graphical representation of linear equations provides a visual aid in understanding the nature of the solutions. For example, if two lines have different slopes, they must intersect at some point, indicating a unique solution. If they have the same slope but different y-intercepts, they are parallel and have no solution. If they have the same slope and the same y-intercept, they are the same line, and every point on the line is a solution.

Methods for Solving Systems of Equations

Several methods exist for solving systems of equations, each with its own strengths and weaknesses. The most common methods include:

• Substitution: Solving one equation for one variable and substituting that expression into the other equation.
• Elimination: Adding or subtracting multiples of the equations to eliminate one variable.
• Graphing: Plotting the equations on a coordinate plane and finding the points of intersection.
• Matrix Methods: Using matrices and linear algebra to solve larger systems of equations.

The choice of method depends on the specific system of equations. Substitution is often effective when one equation can be easily solved for one variable. Elimination is particularly useful when the coefficients of one variable are multiples of each other. Graphing provides a visual representation but can be less accurate for non-integer solutions. Matrix methods are more suitable for complex systems with many variables. Each method has its advantages and limitations, and the ability to choose the most appropriate method is crucial for efficient problem-solving.

Solving the System 6x + 3y = 9 and 3x + 3y = 3 by Elimination

Now, let's tackle the specific system of equations: 6x + 3y = 9 and 3x + 3y = 3. We'll employ the elimination method to solve for 'x.' This method hinges on the idea of manipulating the equations to eliminate one variable, making it easier to solve for the remaining variable. The key is to find a way to add or subtract the equations such that one of the variables cancels out. In this case, the 'y' terms have the same coefficient (3), which makes elimination straightforward.

Step 1: Identifying the Variable to Eliminate

In our system, the 'y' variable is the easiest to eliminate because both equations have the same coefficient (3) for 'y.' This means we can directly subtract one equation from the other to eliminate 'y.' If the coefficients were different, we would need to multiply one or both equations by a constant to make the coefficients match. For example, if the equations were 2x + y = 5 and x - 2y = 1, we could multiply the first equation by 2 to make the coefficients of 'y' equal in magnitude but opposite in sign (2y and -2y). This would allow us to eliminate 'y' by adding the equations. The goal is always to find the most efficient way to make the coefficients match, minimizing the number of steps involved.

Step 2: Eliminating the Variable

To eliminate 'y,' we subtract the second equation (3x + 3y = 3) from the first equation (6x + 3y = 9):

(6x + 3y) - (3x + 3y) = 9 - 3

This simplifies to:

3x = 6

By subtracting the equations, we have successfully eliminated 'y,' leaving us with a simple equation involving only 'x.' This is the core principle of the elimination method – reducing the system to a single equation with a single unknown. The resulting equation is much easier to solve, allowing us to isolate the variable and find its value. The elimination step transforms the original system into a more manageable form, making the solution process more efficient.

Step 3: Solving for x

Now we solve the simplified equation, 3x = 6, for 'x.' Dividing both sides by 3, we get:

x = 2

Thus, the value of 'x' in this system of equations is 2. This is a critical step, as we have now found the value of one of the variables. However, it's important to remember that we have only solved for one variable so far. To fully solve the system, we would also need to find the value of 'y.' In this particular question, we are only asked to find the value of 'x,' so we have reached our desired solution. In other situations, we would proceed by substituting the value of 'x' back into one of the original equations to solve for 'y.'

Importance of Verifying the Solution

After finding a solution, it's crucial to verify its accuracy. This step involves substituting the obtained value(s) back into the original equations to ensure they hold true. For our solution, x = 2, let's verify:

• For the first equation, 6x + 3y = 9: 6(2) + 3y = 9, which simplifies to 12 + 3y = 9.
• For the second equation, 3x + 3y = 3: 3(2) + 3y = 3, which simplifies to 6 + 3y = 3.

To complete the verification, we would need to solve for 'y' in both equations. If the resulting value of 'y' is the same in both cases, then our solution for 'x' is correct. This verification process is essential for catching any potential errors made during the solution process. It provides confidence in the accuracy of the answer and ensures that the solution truly satisfies the system of equations.

Why Verification is Necessary

Verification is not just a formality; it's a vital step in the problem-solving process. Errors can occur during algebraic manipulations, such as incorrect sign changes or arithmetic mistakes. Substituting the solution back into the original equations acts as a check, revealing any inconsistencies. If the solution does not satisfy both equations, it indicates an error somewhere in the solution process. This allows us to go back and review our steps, identify the mistake, and correct it. Verification is a crucial habit to develop in mathematics, as it ensures the reliability of the results and fosters a deeper understanding of the problem-solving process.

Steps to Verify the Solution

The verification process typically involves the following steps:

1. Substitute: Substitute the values obtained for the variables into the original equations.
2. Simplify: Simplify both sides of each equation.
3. Check for Equality: Verify that the left-hand side equals the right-hand side for each equation.
4. Conclusion: If the equality holds true for all equations, the solution is correct. If not, there is an error in the solution process.

By following these steps, you can confidently verify the accuracy of your solutions and ensure that you have correctly solved the system of equations.

Conclusion: Mastering Systems of Equations

Solving systems of equations is a fundamental skill in mathematics with widespread applications. The elimination method, as demonstrated in this article, provides a powerful tool for tackling these problems. By understanding the underlying principles, applying the steps systematically, and verifying the solution, you can confidently solve a wide range of systems of equations. This skill not only enhances your mathematical abilities but also provides a foundation for tackling more complex problems in various fields. Remember, practice is key to mastering this technique, and the more you work with systems of equations, the more proficient you will become.