What Is The Solution To This System Of Linear Equations?${ \begin{array}{l} x + Y = 4 \ x - Y = 6 \end{array} }$A. (4, 6)B. (6, 4)C. (5, -1)D. (-1, 5)

by ADMIN 152 views

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 explore the solution to a system of linear equations, focusing on the given system:

x+y=4x−y=6{ \begin{array}{l} x + y = 4 \\ x - y = 6 \end{array} }

Understanding the System of Linear Equations

The given system consists of two linear equations with two variables, x and y. The first equation is x + y = 4, and the second equation is x - y = 6. To find the solution, we need to determine the values of x and y that satisfy both equations.

Method 1: Addition Method

One way to solve this system is by using the addition method. This method involves adding the two equations together to eliminate one of the variables. In this case, we can add the two equations to eliminate the variable y.

(x+y)+(x−y)=4+62x=10{ \begin{array}{l} (x + y) + (x - y) = 4 + 6 \\ 2x = 10 \end{array} }

By adding the two equations, we have eliminated the variable y and are left with an equation involving only x. We can now solve for x by dividing both sides of the equation by 2.

2x=10x=5{ \begin{array}{l} 2x = 10 \\ x = 5 \end{array} }

Method 2: Subtraction Method

Another way to solve this system is by using the subtraction method. This method involves subtracting one equation from the other to eliminate one of the variables. In this case, we can subtract the second equation from the first equation to eliminate the variable y.

(x+y)−(x−y)=4−62y=−2{ \begin{array}{l} (x + y) - (x - y) = 4 - 6 \\ 2y = -2 \end{array} }

By subtracting the second equation from the first equation, we have eliminated the variable x and are left with an equation involving only y. We can now solve for y by dividing both sides of the equation by 2.

2y=−2y=−1{ \begin{array}{l} 2y = -2 \\ y = -1 \end{array} }

Finding the Solution

Now that we have found the values of x and y, we can substitute them into one of the original equations to verify the solution. Let's use the first equation x + y = 4.

x+y=45+(−1)=4{ \begin{array}{l} x + y = 4 \\ 5 + (-1) = 4 \end{array} }

By substituting x = 5 and y = -1 into the first equation, we can see that the solution satisfies the equation.

Conclusion

In conclusion, the solution to the system of linear equations is x = 5 and y = -1. This solution satisfies both equations in the system, and it can be verified by substituting the values into one of the original equations.

Final Answer

The final answer is (5, -1).

Discussion

This system of linear equations can be solved using either the addition method or the subtraction method. Both methods involve eliminating one of the variables and solving for the other variable. The solution to the system is x = 5 and y = -1, which can be verified by substituting the values into one of the original equations.

Related Topics

  • Systems of linear equations
  • Addition method
  • Subtraction method
  • Linear equations
  • Algebra
  • Geometry

References

  • [1] "Systems of Linear Equations" by Math Open Reference
  • [2] "Addition Method" by Mathway
  • [3] "Subtraction Method" by Purplemath

Tags

  • Systems of linear equations
  • Addition method
  • Subtraction method
  • Linear equations
  • Algebra
  • Geometry

Categories

  • Mathematics
  • Algebra
  • Geometry

Keywords

  • Systems of linear equations
  • Addition method
  • Subtraction method
  • Linear equations
  • Algebra
  • Geometry

Description

This article provides an explanation of the solution to a system of linear equations using the addition method and the subtraction method. The solution is x = 5 and y = -1, which can be verified by substituting the values into one of the original equations. The article also provides related topics, references, and tags for further learning.

Introduction

Systems of linear equations are a fundamental concept in mathematics, particularly in algebra and geometry. In this article, we will address some of the most frequently asked questions about systems of linear equations.

Q1: What is a system of linear equations?

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.

Q2: How do I solve a system of linear equations?

There are several methods to solve a system of linear equations, including the addition method, subtraction method, and graphing method. The choice of method depends on the type of system and the number of equations.

Q3: What is the addition method?

The addition method involves adding the two equations together to eliminate one of the variables. This method is useful when the coefficients of the variables are the same in both equations.

Q4: What is the subtraction method?

The subtraction method involves subtracting one equation from the other to eliminate one of the variables. This method is useful when the coefficients of the variables are different in both equations.

Q5: How do I determine the number of solutions to a system of linear equations?

The number of solutions to a system of linear equations depends on the type of system. If the system has a unique solution, it means that there is only one set of values that satisfies all the equations. If the system has no solution, it means that there is no set of values that satisfies all the equations. If the system has infinitely many solutions, it means that there are many sets of values that satisfy all the equations.

Q6: What is the difference between a system of linear equations and a system of nonlinear equations?

A system of linear equations involves linear equations, which are equations that can be written in the form ax + by = c, where a, b, and c are constants. A system of nonlinear equations involves nonlinear equations, which are equations that cannot be written in the form ax + by = c.

Q7: How do I graph a system of linear equations?

To graph a system of linear equations, you can plot the equations on a coordinate plane and find the point of intersection. The point of intersection represents the solution to the system.

Q8: What is the importance of systems of linear equations in real-life applications?

Systems of linear equations have many real-life applications, including physics, engineering, economics, and computer science. They are used to model real-world problems and make predictions about the behavior of systems.

Q9: Can I use a calculator to solve a system of linear equations?

Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q10: How do I check my solution to a system of linear equations?

To check your solution to a system of linear equations, you can substitute the values into one of the original equations and verify that it is true.

Conclusion

In conclusion, systems of linear equations are a fundamental concept in mathematics, and they have many real-life applications. By understanding the different methods for solving systems of linear equations, you can apply them to a wide range of problems.

Final Answer

The final answer is (5, -1).

Discussion

This article provides an explanation of the solution to a system of linear equations using the addition method and the subtraction method. The solution is x = 5 and y = -1, which can be verified by substituting the values into one of the original equations. The article also provides related topics, references, and tags for further learning.

Related Topics

  • Systems of linear equations
  • Addition method
  • Subtraction method
  • Linear equations
  • Algebra
  • Geometry

References

  • [1] "Systems of Linear Equations" by Math Open Reference
  • [2] "Addition Method" by Mathway
  • [3] "Subtraction Method" by Purplemath

Tags

  • Systems of linear equations
  • Addition method
  • Subtraction method
  • Linear equations
  • Algebra
  • Geometry

Categories

  • Mathematics
  • Algebra
  • Geometry

Keywords

  • Systems of linear equations
  • Addition method
  • Subtraction method
  • Linear equations
  • Algebra
  • Geometry

Description

This article provides an explanation of the solution to a system of linear equations using the addition method and the subtraction method. The solution is x = 5 and y = -1, which can be verified by substituting the values into one of the original equations. The article also provides related topics, references, and tags for further learning.