Why System B is Equivalent to System A

Oct 13, 2025 by talynt.org 88 views

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. These systems can be linear or non-linear, and they can have one or more solutions. In this article, we will discuss a system of equations and its solution, and we will explore how to get another system of equations from the given one.

System A

The system of equations given below is System A.

System A

{ \begin{aligned} 5x - y & = -11 \\ 3x - 2y & = -8 \\ \text{Solution: } & (-2, 1) \end{aligned} \}

In this system, we have two equations with two variables, x and y. The first equation is 5x - y = -11, and the second equation is 3x - 2y = -8. The solution to this system is given as (-2, 1), which means that when x = -2 and y = 1, both equations are satisfied.

System B

To get System B, we need to modify the second equation in System A. We can do this by multiplying the first equation by 2 and then adding it to the second equation.

Modified Second Equation

Let's multiply the first equation by 2:

{ \begin{aligned} 10x - 2y & = -22 \end{aligned} \}

Now, let's add this equation to the second equation:

{ \begin{aligned} (10x - 2y) + (3x - 2y) & = -22 + (-8) \\ 13x - 4y & = -30 \end{aligned} \}

This is the modified second equation.

System B

Now, let's write down System B using the modified second equation.

System B

{ \begin{aligned} 5x - y & = -11 \\ 13x - 4y & = -30 \\ \text{Solution: } & (-2, 1) \end{aligned} \}

In System B, we have the same first equation as in System A, but the second equation is different. The solution to this system is still (-2, 1).

Why System B is Equivalent to System A

System B is equivalent to System A because they have the same solution. This means that the two systems are consistent, and the solution to one system is also a solution to the other system.

To see why this is the case, let's substitute the solution (-2, 1) into the second equation of System B:

{ \begin{aligned} 13(-2) - 4(1) & = -30 \\ -26 - 4 & = -30 \\ -30 & = -30 \end{aligned} \}

As we can see, the solution (-2, 1) satisfies the second equation of System B. This means that System B is consistent, and the solution to System A is also a solution to System B.

Conclusion

In this article, we discussed a system of equations and its solution, and we explored how to get another system of equations from the given one. We saw that System B is equivalent to System A because they have the same solution. This means that the two systems are consistent, and the solution to one system is also a solution to the other system.

Key Takeaways

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
System A is a system of equations with two equations and two variables, and its solution is given as (-2, 1).
System B is a modified version of System A, and it has the same solution as System A.
System B is equivalent to System A because they have the same solution.

Introduction

In our previous article, we discussed a system of equations and its solution, and we explored how to get another system of equations from the given one. In this article, we will answer some frequently asked questions about systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: How do I know if a system of equations has a solution?

A: To determine if a system of equations has a solution, you can use the following methods:

Graphing: Graph the equations on a coordinate plane and see if they intersect.
Substitution: Substitute the expression for one variable into the other equation and solve for the remaining variable.
Elimination: Add or subtract the equations to eliminate one variable and solve for the remaining variable.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the following methods:

Substitution: Substitute the expression for one variable into the other equation and solve for the remaining variable.
Elimination: Add or subtract the equations to eliminate one variable and solve for the remaining variable.
Graphing: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the difference between a linear and a non-linear system of equations?

A: A linear system of equations is a system of equations where the variables are raised to the power of 1. A non-linear system of equations is a system of equations where the variables are raised to a power other than 1.

Q: Can a system of equations have more than one solution?

A: Yes, a system of equations can have more than one solution. This is known as an inconsistent system of equations.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution. This is known as an inconsistent system of 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 use the following methods:

Graphing: Graph the equations on a coordinate plane and see if they intersect.
Substitution: Substitute the expression for one variable into the other equation and solve for the remaining variable.
Elimination: Add or subtract the equations to eliminate one variable and solve for the remaining variable.

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

A: A dependent system of equations is a system of equations where the equations are linearly dependent, meaning that one equation can be written as a multiple of the other equation. An independent system of equations is a system of equations where the equations are linearly independent, meaning that one equation cannot be written as a multiple of the other equation.

Q: Can a system of equations be both dependent and independent?

A: No, a system of equations cannot be both dependent and independent. A system of equations is either dependent or independent, but not both.

Conclusion

In this article, we answered some frequently asked questions about systems of equations. We discussed the definition of a system of equations, how to determine if a system of equations has a solution, and how to solve a system of equations. We also discussed the difference between a linear and a non-linear system of equations, and how to determine if a system of equations is consistent or inconsistent.

Key Takeaways

A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
A system of equations can have one or more solutions.
A system of equations can be linear or non-linear.
A system of equations can be consistent or inconsistent.
A system of equations can be dependent or independent.