Understanding the Linear System

Oct 13, 2025 by pynkocean.com 119 views

Introduction

In mathematics, a linear system is a set of two or more linear equations that are to be solved simultaneously. These equations are typically represented in the form of ax + by = c, where a, b, and c are constants, and x and y are the variables. Solving a linear system involves finding the values of x and y that satisfy all the equations in the system. In this article, we will explore how to solve a linear system and determine which ordered pair is a solution to the given system.

Understanding the Linear System

The given linear system consists of two equations:

2x + y = 5
x - y = 13

To solve this system, we need to find the values of x and y that satisfy both equations. We can use various methods to solve a linear system, including substitution, elimination, and graphing. In this case, we will use the substitution method.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Let's solve the second equation for x:

x = 13 + y

Now, substitute this expression for x into the first equation:

2(13 + y) + y = 5

Expand and simplify the equation:

26 + 2y + y = 5

Combine like terms:

26 + 3y = 5

Subtract 26 from both sides:

3y = -21

Divide both sides by 3:

y = -7

Now that we have found the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:

x - (-7) = 13

Simplify the equation:

x + 7 = 13

Subtract 7 from both sides:

x = 6

Checking the Solution

To verify that the ordered pair (6, -7) is a solution to the linear system, we need to check if it satisfies both equations. Let's plug in x = 6 and y = -7 into both equations:

2x + y = 5 2(6) + (-7) = 5 12 - 7 = 5 5 = 5 (True)
x - y = 13 6 - (-7) = 13 6 + 7 = 13 13 = 13 (True)

Since the ordered pair (6, -7) satisfies both equations, it is indeed a solution to the linear system.

Conclusion

In this article, we have explored how to solve a linear system using the substitution method. We have also determined which ordered pair is a solution to the given system. The correct answer is (6, -7). This solution satisfies both equations in the system, making it the correct choice.

Frequently Asked Questions

What is a linear system?
How do you solve a linear system?
What is the substitution method?
How do you check if an ordered pair is a solution to a linear system?

Final Answer

The final answer is (6, -7).

Introduction

In our previous article, we explored how to solve a linear system using the substitution method and determined which ordered pair is a solution to the given system. In this article, we will answer some frequently asked questions related to solving linear systems.

Q&A

Q: What is a linear system?

A: A linear system is a set of two or more linear equations that are to be solved simultaneously. These equations are typically represented in the form of ax + by = c, where a, b, and c are constants, and x and y are the variables.

Q: How do you solve a linear system?

A: There are several methods to solve a linear system, including substitution, elimination, and graphing. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method of solving a linear system by solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one of the equations is already solved for one variable.

Q: How do you check if an ordered pair is a solution to a linear system?

A: To check if an ordered pair is a solution to a linear system, plug in the values of x and y into both equations and check if the equations are true. If the ordered pair satisfies both equations, it is a solution to the linear system.

Q: What is the difference between a linear system and a nonlinear system?

A: A linear system is a set of linear equations, while a nonlinear system is a set of nonlinear equations. Nonlinear equations are equations that are not in the form of ax + by = c, where a, b, and c are constants, and x and y are the variables.

Q: Can a linear system have more than two equations?

A: Yes, a linear system can have more than two equations. However, it is typically easier to solve a linear system with two equations.

Q: How do you solve a linear system with three or more equations?

A: To solve a linear system with three or more equations, use the substitution or elimination method to reduce the system to two equations, and then solve the resulting system.

Q: What is the importance of solving linear systems?

A: Solving linear systems is important in many real-world applications, such as physics, engineering, and economics. It is used to model and solve problems involving multiple variables and equations.

Conclusion

In this article, we have answered some frequently asked questions related to solving linear systems. We have covered topics such as the definition of a linear system, the substitution method, and how to check if an ordered pair is a solution to a linear system. We hope that this article has provided you with a better understanding of solving linear systems.

Final Answer

The final answer is (6, -7).