Finding the Equivalent Term

Oct 13, 2025 by pynkocean.com 24 views

Introduction

In mathematics, solving linear systems is a fundamental concept that involves finding the solution to a system of linear equations. A linear system is a set of equations in which each equation is linear, meaning it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. In this article, we will explore how to solve linear systems using a table, and we will focus on finding the term that will make the systems equivalent.

Understanding the Problem

The problem presents two linear systems, System A and System B, as shown in the table below.

System A	System B
7x + 4y = -4	7x + 4y = -4
5x + 8y = 28	?

The goal is to find the term that will make the two systems equivalent. In other words, we need to find the value that will make the two systems have the same solution.

Analyzing the Systems

To solve this problem, we need to analyze the two systems and identify the term that will make them equivalent. Let's start by examining the first equation in each system.

In System A, the first equation is 7x + 4y = -4.
In System B, the first equation is also 7x + 4y = -4.

Since the first equations are the same, we can conclude that the systems will be equivalent if the second equations are also the same.

Finding the Equivalent Term

To find the equivalent term, we need to examine the second equation in System B. The second equation in System B is missing, so we need to find the value that will make it equivalent to the second equation in System A, which is 5x + 8y = 28.

Let's analyze the second equation in System A.

The coefficient of x is 5.
The coefficient of y is 8.
The constant term is 28.

To make the second equation in System B equivalent to the second equation in System A, we need to find the value that will make the coefficients and the constant term the same.

The Correct Answer

The correct answer is -14x.

To see why, let's substitute -14x into the second equation in System B.

The new equation becomes 7x + 4y = -4.
The coefficient of x is still 7.
The coefficient of y is still 4.
The constant term is still -4.

Now, let's examine the second equation in System A.

The coefficient of x is 5.
The coefficient of y is 8.
The constant term is 28.

Since the coefficients and the constant term are the same, we can conclude that the two systems are equivalent.

Conclusion

In this article, we explored how to solve linear systems using a table. We analyzed the two systems and identified the term that will make them equivalent. The correct answer is -14x. We hope this article has provided a clear and concise explanation of how to solve linear systems and find the equivalent term.

Frequently Asked Questions

Q: What is a linear system?

A: A linear system is a set of equations in which each equation is linear, meaning it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I find the equivalent term in a linear system?

A: To find the equivalent term, you need to examine the equations in the system and identify the term that will make the coefficients and the constant term the same.

Q: What is the correct answer to the problem?

A: The correct answer is -14x.

Q: How do I know if two linear systems are equivalent?

A: Two linear systems are equivalent if the coefficients and the constant term are the same in both systems.

References

Linear Systems
Solving Linear Systems
Linear Algebra
Frequently Asked Questions: Linear Systems =============================================

Q: What is a linear system?

A: A linear system is a set of equations in which each equation is linear, meaning it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I know if a system of equations is linear?

A: To determine if a system of equations is linear, you need to examine each equation and check if it can be written in the form ax + by = c. If all the equations can be written in this form, then the system is linear.

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

A: A linear system is a set of equations in which each equation is linear, meaning it can be written in the form ax + by = c. A nonlinear system, on the other hand, is a set of equations in which at least one equation is nonlinear, meaning it cannot be written in the form ax + by = c.

Q: How do I solve a linear system?

A: To solve a linear system, you can use various methods such as substitution, elimination, or graphing. The method you choose will depend on the specific system and the variables involved.

Q: What is the difference between a dependent and an independent system?

A: A dependent system is a system of equations in which the equations are not independent, meaning that one equation can be written as a multiple of the other. An independent system, on the other hand, is a system of equations in which the equations are independent, meaning that none of the equations can be written as a multiple of the others.

Q: How do I determine if a system of equations is dependent or independent?

A: To determine if a system of equations is dependent or independent, you can examine the equations and check if one equation can be written as a multiple of the other. If one equation can be written as a multiple of the other, then the system is dependent. If none of the equations can be written as a multiple of the others, then the system is independent.

Q: What is the difference between a consistent and an inconsistent system?

A: A consistent system is a system of equations in which the equations have a solution, meaning that there is at least one value of x and y that satisfies all the equations. An inconsistent system, on the other hand, is a system of equations in which the equations have no solution, meaning that there is no value of x and y that satisfies all the equations.

Q: How do I determine if a system of equations is consistent or inconsistent?

A: To determine if a system of equations is consistent or inconsistent, you can examine the equations and check if they have a solution. If the equations have a solution, then the system is consistent. If the equations have no solution, then the system is inconsistent.

Q: What is the difference between a unique and a non-unique solution?

A: A unique solution is a solution in which there is only one value of x and y that satisfies all the equations. A non-unique solution, on the other hand, is a solution in which there are multiple values of x and y that satisfy all the equations.

Q: How do I determine if a system of equations has a unique or non-unique solution?

A: To determine if a system of equations has a unique or non-unique solution, you can examine the equations and check if they have a unique solution. If the equations have a unique solution, then the system has a unique solution. If the equations have multiple solutions, then the system has a non-unique solution.

Q: What is the difference between a linear equation and a linear inequality?

A: A linear equation is an equation in which the variables are raised to the power of 1, and the coefficients are constants. A linear inequality, on the other hand, is an inequality in which the variables are raised to the power of 1, and the coefficients are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can use various methods such as graphing, substitution, or elimination. The method you choose will depend on the specific inequality and the variables involved.

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

A: A linear system is a set of equations in which each equation is linear, meaning it can be written in the form ax + by = c. A linear programming problem, on the other hand, is a problem in which the objective function and the constraints are linear.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, you can use various methods such as the simplex method, the dual simplex method, or the interior-point method. The method you choose will depend on the specific problem and the variables involved.

Conclusion

In this article, we have answered some of the most frequently asked questions about linear systems. We have covered topics such as the definition of a linear system, how to determine if a system is linear or nonlinear, how to solve a linear system, and how to determine if a system has a unique or non-unique solution. We hope this article has provided a clear and concise explanation of these concepts and has helped you to better understand linear systems.