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
Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. They consist of two or more linear equations that involve variables, and the goal is to find the values of these variables that satisfy all the equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables.
Understanding the Problem
We are given a system of two linear equations:
Our objective is to find the values of x and y that satisfy both equations.
Method 1: Substitution Method
One way to solve this system is by using the substitution method. We can solve one equation for one variable and then substitute that expression into the other equation.
Let's solve the first equation for y:
Subtracting 2x from both sides gives us:
Now, we can substitute this expression for y into the second equation:
Expanding and simplifying the equation, we get:
Combine like terms:
Add 1 to both sides:
Divide both sides by 5:
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 x = -1, we get:
Simplifying the equation, we get:
Adding 2 to both sides:
Therefore, the solution to the system of linear equations is x = -1 and y = 3.
Method 2: Elimination Method
Another way to solve this system is by using the elimination method. We can multiply both equations by necessary multiples such that the coefficients of y's in both equations are the same:
Multiply the first equation by 1 and the second equation by 1:
Add both equations to eliminate the y variable:
Combine like terms:
Divide both sides by 5:
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 x = -1, we get:
Simplifying the equation, we get:
Adding 2 to both sides:
Therefore, the solution to the system of linear equations is x = -1 and y = 3.
Conclusion
In this article, we have solved a system of two linear equations with two variables using the substitution method and the elimination method. We have found that the solution to the system is x = -1 and y = 3. This solution satisfies both equations, and it is the only solution that does so.
Final Answer
The final answer 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 variables. The goal is to find the values of these variables that satisfy all the equations simultaneously.
Q: How do I know if a system of linear equations has a solution?
A: A system of linear equations has a solution if and only if the two equations are not parallel. If the equations are parallel, then there is no solution. If the equations intersect, then there is a unique solution.
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.
Q: What is the substitution method?
A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
Q: What is the elimination method?
A: The elimination method involves multiplying both equations by necessary multiples such that the coefficients of y's in both equations are the same, and then adding both equations to eliminate the y variable.
Q: How do I choose which method to use?
A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, then the elimination method is a good choice. If the coefficients of one variable are different in both equations, then the substitution method is a good choice.
Q: What if I have a system of linear equations with three or more variables?
A: If you have a system of linear equations with three or more variables, then you can use the same methods as before, but you may need to use more complex techniques such as Gaussian elimination or matrix operations.
Q: Can I use a calculator or computer to solve a system of linear equations?
A: Yes, you can use a calculator or computer to solve a system of linear equations. Many calculators and computer programs have built-in functions for solving systems of linear equations.
Q: How do I check if my solution is correct?
A: To check if your solution is correct, you can plug the values of the variables back into both equations and see if they are true. If they are true, then your solution is correct.
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, then the equations are parallel and there is no intersection point.
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, then the equations are the same and there are infinitely many points that satisfy both equations.
Q: Can I use systems of linear equations to model real-world problems?
A: Yes, you can use systems of linear equations to model real-world problems. Many problems in science, engineering, economics, and other fields can be represented as systems of linear equations.
Q: How do I apply systems of linear equations to real-world problems?
A: To apply systems of linear equations to real-world problems, you need to identify the variables and the equations that represent the problem. Then, you can use the methods for solving systems of linear equations to find the solution.
Q: What are some common applications of systems of linear equations?
A: Some common applications of systems of linear equations include:
- Modeling population growth
- Solving problems in physics and engineering
- Analyzing data in economics and finance
- Solving problems in computer science and programming
- Modeling real-world problems in science and mathematics
Conclusion
In this article, we have answered some 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 we have provided examples and explanations to help you understand how to apply these methods to real-world problems.