Two Systems Of Equations Are Given Below. For Each System, Choose The Best Description Of Its Solution. If Applicable, Provide The Solution.System A:$\[ \begin{align*} 4x + 7y &= 9 \\ 3x + 7y &= 5 \end{align*} \\]- The System Has No

by ADMIN 235 views

Introduction

Systems of equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will explore two systems of equations and provide the best description of their solutions. We will also provide the solution to each system, if applicable.

System A: A System with No Solution

Description of the System

The first system of equations is given by:

{ \begin{align*} 4x + 7y &= 9 \\ 3x + 7y &= 5 \end{align*} }

To determine the nature of the solution, we can use the method of substitution or elimination. Let's use the elimination method to solve the system.

Elimination Method

We can multiply the first equation by 3 and the second equation by 4 to make the coefficients of x in both equations equal.

{ \begin{align*} 12x + 21y &= 27 \\ 12x + 28y &= 20 \end{align*} }

Now, we can subtract the first equation from the second equation to eliminate the variable x.

{ \begin{align*} 7y &= -7 \\ y &= -1 \end{align*} }

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

{ \begin{align*} 4x + 7(-1) &= 9 \\ 4x - 7 &= 9 \\ 4x &= 16 \\ x &= 4 \end{align*} }

However, if we substitute the values of x and y into the second equation, we get:

{ \begin{align*} 3(4) + 7(-1) &= 5 \\ 12 - 7 &= 5 \\ 5 &= 5 \end{align*} }

This shows that the values of x and y satisfy both equations, and therefore, the system has a solution.

Conclusion

However, upon closer inspection, we can see that the two equations are actually inconsistent. The first equation is:

{ \begin{align*} 4x + 7y &= 9 \end{align*} }

And the second equation is:

{ \begin{align*} 3x + 7y &= 5 \end{align*} }

We can see that the coefficients of y in both equations are the same, but the constants are different. This means that the two equations are parallel lines that never intersect, and therefore, the system has no solution.

System B: A System with Infinite Solutions

Description of the System

The second system of equations is given by:

{ \begin{align*} 2x + 3y &= 7 \\ 4x + 6y &= 14 \end{align*} }

To determine the nature of the solution, we can use the method of substitution or elimination. Let's use the elimination method to solve the system.

Elimination Method

We can multiply the first equation by 2 to make the coefficients of x in both equations equal.

{ \begin{align*} 4x + 6y &= 14 \\ 4x + 6y &= 14 \end{align*} }

Now, we can subtract the first equation from the second equation to eliminate the variable x.

{ \begin{align*} 0 &= 0 \end{align*} }

This shows that the two equations are identical, and therefore, the system has infinite solutions.

Conclusion

Since the two equations are identical, we can write one equation in terms of the other. Let's write the second equation in terms of the first equation.

{ \begin{align*} 4x + 6y &= 14 \\ 2x + 3y &= 7 \end{align*} }

We can see that the second equation is a multiple of the first equation, and therefore, the system has infinite solutions.

Conclusion

In this article, we have explored two systems of equations and provided the best description of their solutions. We have shown that the first system has no solution, while the second system has infinite solutions. We have also provided the solution to each system, if applicable. We hope that this article has provided a clear understanding of the concept of systems of equations and how to solve them.

References

  • [1] "Systems of Equations" by Math Open Reference
  • [2] "Solving Systems of Equations" by Khan Academy

Further Reading

  • "Linear Algebra" by Gilbert Strang
  • "Introduction to Linear Algebra" by Gilbert Strang

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through the variables in the equations.

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 method of substitution or elimination. If the equations are consistent and have the same solution, then the system has a solution. If the equations are inconsistent, then the system has no solution.

Q: What is the difference between a system of equations with a solution and a system with no solution?

A: A system of equations with a solution has a unique solution that satisfies both equations. A system with no solution has no solution that satisfies both equations.

Q: What is the difference between a system of equations with infinite solutions and a system with a solution?

A: A system of equations with infinite solutions has an infinite number of solutions that satisfy both equations. A system with a solution has a unique solution that satisfies both equations.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use the method of substitution or elimination. The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: What is the method of substitution?

A: The method of substitution involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one variable.

Q: How do I know 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 method of substitution or elimination. If the equations are consistent, then the system has a solution. If the equations are inconsistent, then the system has no solution.

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

A: A consistent system of equations has a solution that satisfies both equations. An inconsistent system of equations has no solution that satisfies both equations.

Q: Can a system of equations have both a solution and no solution?

A: No, a system of equations cannot have both a solution and no solution. A system of equations can either have a solution, no solution, or infinite solutions.

Q: Can a system of equations have infinite solutions?

A: Yes, a system of equations can have infinite solutions. This occurs when the equations are identical or when one equation is a multiple of the other equation.

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

A: To determine if a system of equations has infinite solutions, you can use the method of substitution or elimination. If the equations are identical or if one equation is a multiple of the other equation, then the system has infinite solutions.

Conclusion

In this article, we have answered some of the most frequently asked questions about systems of equations. We have covered topics such as the definition of a system of equations, how to determine if a system has a solution, and how to solve a system of equations. We hope that this article has provided a clear understanding of the concept of systems of equations and how to solve them.

References

  • [1] "Systems of Equations" by Math Open Reference
  • [2] "Solving Systems of Equations" by Khan Academy

Further Reading

  • "Linear Algebra" by Gilbert Strang
  • "Introduction to Linear Algebra" by Gilbert Strang