What Is The Solution To This System Of Linear Equations?${ \begin{array}{l} 2x + Y = 1 \ 3x - Y = -6 \end{array} }$A. { (-1, 3)$}$B. { (1, -1)$}$C. { (2, 3)$}$D. { (5, 0)$}$
Introduction to Systems of Linear Equations
A system of linear equations is a set of two or more linear equations that involve the same set of variables. In this article, we will focus on solving a system of two linear equations with two variables. The system of linear equations we will be working with is:
{ \begin{array}{l} 2x + y = 1 \\ 3x - y = -6 \end{array} \}
Understanding the Problem
To solve this system of linear equations, we need to find the values of the variables x and y that satisfy both equations simultaneously. In other words, we need to find the point of intersection of the two lines represented by the equations.
Method 1: Substitution Method
One way to solve this system of linear equations is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.
Let's start by solving the first equation for y:
Subtracting 2x from both sides gives us:
Now, substitute this expression for y into the second equation:
Substituting into the second equation gives us:
Expanding and simplifying the equation gives us:
Combine like terms:
Adding 1 to both sides gives us:
Dividing both sides by 5 gives us:
Now that we have found 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:
Substituting into the first equation gives us:
Simplifying the equation gives us:
Adding 2 to both sides gives us:
Therefore, the solution to the system of linear equations is:
Method 2: Elimination Method
Another way to solve this system of linear equations is by using the elimination method. This method involves adding or subtracting the two equations to eliminate one of the variables.
Let's start by adding the two equations:
Adding the two equations gives us:
Dividing both sides by 5 gives us:
Now that we have found 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:
Substituting into the first equation gives us:
Simplifying the equation gives us:
Adding 2 to both sides gives us:
Therefore, the solution to the system of linear equations is:
Conclusion
In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods have led us to the same solution:
This solution represents the point of intersection of the two lines represented by the equations. We can verify this solution by plugging it back into both equations to make sure it satisfies both equations simultaneously.
Final Answer
The final answer to the system of linear equations is:
This is the correct solution to the system of linear equations.
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. In this article, we have focused on solving a system of two linear equations with two variables.
Q: What are the two main methods for solving systems of linear equations?
A: The two main methods for solving systems of linear equations are the substitution method and the elimination method. The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the two equations to eliminate one of the variables.
Q: How do I know which method to use?
A: The choice of method depends on the specific system of linear equations you are working with. If one of the equations is already solved for one variable, the substitution method may be easier to use. If the coefficients of the variables are the same in both equations, the elimination method may be easier to use.
Q: What if I have a system of linear equations with more than two variables?
A: If you have a system of linear equations with more than two variables, you can use the same methods we have discussed, but you will need to use more complex algebraic manipulations. You may also need to use other techniques, such as graphing or matrix methods.
Q: How do I verify that my solution is correct?
A: To verify that your solution is correct, you can plug it back into both equations to make sure it satisfies both equations simultaneously. If the solution satisfies both equations, then it is the correct solution.
Q: What if I have a system of linear equations with no solution?
A: If you have a system of linear equations with no solution, it means that the two lines represented by the equations do not intersect. This can happen if the equations are inconsistent, meaning that they cannot be true at the same time.
Q: What if I have a system of linear equations with infinitely many solutions?
A: If you have a system of linear equations with infinitely many solutions, it means that the two lines represented by the equations are the same line. This can happen if the equations are dependent, meaning that one equation is a multiple of the other.
Q: Can I use technology to solve systems of linear equations?
A: Yes, you can use technology, such as graphing calculators or computer software, to solve systems of linear equations. These tools can help you visualize the problem and find the solution more easily.
Q: Are there any other methods for solving systems of linear equations?
A: Yes, there are other methods for solving systems of linear equations, such as the matrix method and the Gaussian elimination method. These methods involve using matrices to represent the system of linear equations and then performing row operations to solve the system.
Q: Can I apply the methods for solving systems of linear equations to other types of equations?
A: Yes, the methods for solving systems of linear equations can be applied to other types of equations, such as quadratic equations and polynomial equations. However, the specific techniques and methods may vary depending on the type of equation.
Conclusion
In this article, we have answered some of the most frequently asked questions about solving systems of linear equations. We have discussed the two main methods for solving systems of linear equations, the substitution method and the elimination method, and provided examples of how to use these methods. We have also discussed how to verify that a solution is correct and how to handle systems of linear equations with no solution or infinitely many solutions.