The Matrix Equation Below Represents A Two-variable Linear System. Are There Solutions? Explain.$\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6

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Introduction

In this article, we will explore the matrix equation that represents a two-variable linear system. The equation is given by [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right]. Our goal is to determine if there are solutions to this system and explain the reasoning behind our answer.

What is a Linear System?

A linear system is a set of equations in which the unknowns are related by linear equations. In other words, each equation in the system is a linear combination of the unknowns. The general form of a linear system is:

a11x1+a12x2+โ‹ฏ+a1nxn=b1a21x1+a22x2+โ‹ฏ+a2nxn=b2โ‹ฎam1x1+am2x2+โ‹ฏ+amnxn=bm\begin{aligned} a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} &= b_{1} \\ a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} &= b_{2} \\ & \vdots \\ a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} &= b_{m} \end{aligned}

where aija_{ij} are constants, xix_{i} are the unknowns, and bib_{i} are the constants on the right-hand side.

The Matrix Equation

The matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right] can be written in the form of a linear system as follows:

3x+2y=66x+4y=0\begin{aligned} 3x + 2y &= 6 \\ 6x + 4y &= 0 \end{aligned}

Solving the System

To determine if there are solutions to this system, we can use the method of substitution or elimination. Let's use the elimination method.

First, we can multiply the first equation by 2 and the second equation by 1 to get:

6x+4y=126x+4y=0\begin{aligned} 6x + 4y &= 12 \\ 6x + 4y &= 0 \end{aligned}

Now, we can subtract the second equation from the first equation to get:

0=12\begin{aligned} 0 &= 12 \end{aligned}

This is a contradiction, which means that the system has no solution.

Conclusion

In conclusion, the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right] represents a two-variable linear system that has no solution. This is because the system is inconsistent, meaning that it is impossible to find values of xx and yy that satisfy both equations.

Why is this System Inconsistent?

The system is inconsistent because the two equations are linearly dependent. In other words, one equation is a multiple of the other. Specifically, the second equation is twice the first equation. This means that the two equations are essentially the same, and there is no solution that can satisfy both equations.

What does this Mean?

This means that the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right] has no solution. This is because the system is inconsistent, and there is no value of xx and yy that can satisfy both equations.

Implications

The fact that the system has no solution has important implications. For example, if we were to use this system to model a real-world problem, we would need to revisit our assumptions and adjust the system accordingly. This could involve changing the coefficients of the equations or adding new equations to the system.

Conclusion

Introduction

In our previous article, we explored the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right] and determined that it represents a two-variable linear system with no solution. In this article, we will answer some frequently asked questions about this system.

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

A: A consistent system is one that has at least one solution, while an inconsistent system is one that has no solution. In the case of the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right], the system is inconsistent because it is impossible to find values of xx and yy that satisfy both equations.

Q: How can I determine if a system is consistent or inconsistent?

A: To determine if a system is consistent or inconsistent, you can use the method of substitution or elimination. If you can find values of the unknowns that satisfy all the equations, then the system is consistent. If you cannot find such values, then the system is inconsistent.

Q: What is the significance of the determinant in a matrix equation?

A: The determinant of a matrix is a scalar value that can be used to determine if the matrix is invertible. If the determinant is zero, then the matrix is not invertible, and the system is inconsistent. In the case of the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right], the determinant is zero, which means that the system is inconsistent.

Q: Can I use the matrix equation to model a real-world problem?

A: While the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right] may not be useful for modeling a real-world problem, it can still be used to illustrate important concepts in linear algebra. For example, it can be used to demonstrate the concept of linear dependence and the importance of the determinant in determining the consistency of a system.

Q: How can I modify the matrix equation to make it consistent?

A: To modify the matrix equation to make it consistent, you can change the coefficients of the equations or add new equations to the system. For example, you could change the second equation to 6x+4y=126x + 4y = 12, which would make the system consistent.

Q: What are some common mistakes to avoid when working with matrix equations?

A: Some common mistakes to avoid when working with matrix equations include:

  • Not checking the consistency of the system before attempting to solve it
  • Not using the correct method for solving the system (e.g. substitution or elimination)
  • Not checking the determinant of the matrix before attempting to invert it
  • Not being careful when multiplying matrices or vectors

Conclusion

In conclusion, the matrix equation [3264][xy]=[6]\left[\begin{array}{ll} 3 & 2 \\ 6 & 4 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{c} 6 \end{array}\right] represents a two-variable linear system with no solution. By understanding the concepts of consistency and inconsistency, and by being careful when working with matrix equations, you can avoid common mistakes and successfully solve linear systems.

Additional Resources

For more information on linear algebra and matrix equations, see the following resources: