Solve The System Of Equations Using Cramer's Rule.${ \begin{cases} 5x + 4y = -2 \ 2x - Y = 0 \end{cases} }$Find The Values Of { X$}$ And { Y$}$.

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Introduction

Cramer's Rule is a method used to solve systems of linear equations. It is a powerful tool for finding the values of variables in a system of equations. In this article, we will use Cramer's Rule to solve a system of two linear equations with two variables.

What is Cramer's Rule?

Cramer's Rule is a method for solving systems of linear equations. It is based on the concept of determinants, which are used to find the solution to a system of equations. The rule states that if we have a system of n linear equations with n variables, we can find the values of the variables by using the following formula:

xi=DiDx_i = \frac{D_i}{D}

where DiD_i is the determinant of the matrix formed by replacing the ithi^{th} column of the coefficient matrix with the constant matrix, and DD is the determinant of the coefficient matrix.

The System of Equations

The system of equations we will be using to demonstrate Cramer's Rule is:

{5x+4y=−22x−y=0\begin{cases} 5x + 4y = -2 \\ 2x - y = 0 \end{cases}

Step 1: Find the Determinant of the Coefficient Matrix

The coefficient matrix is:

[542−1]\begin{bmatrix} 5 & 4 \\ 2 & -1 \end{bmatrix}

To find the determinant of this matrix, we can use the formula:

D=ad−bcD = ad - bc

where aa, bb, cc, and dd are the elements of the matrix.

Plugging in the values, we get:

D=(5)(−1)−(4)(2)D = (5)(-1) - (4)(2)

D=−5−8D = -5 - 8

D=−13D = -13

Step 2: Find the Determinants of the Matrices DxD_x and DyD_y

To find the determinants of the matrices DxD_x and DyD_y, we need to replace the xx and yy columns of the coefficient matrix with the constant matrix.

The matrix DxD_x is:

[−240−1]\begin{bmatrix} -2 & 4 \\ 0 & -1 \end{bmatrix}

The determinant of this matrix is:

Dx=(−2)(−1)−(4)(0)D_x = (-2)(-1) - (4)(0)

Dx=2D_x = 2

The matrix DyD_y is:

[5−220]\begin{bmatrix} 5 & -2 \\ 2 & 0 \end{bmatrix}

The determinant of this matrix is:

Dy=(5)(0)−(−2)(2)D_y = (5)(0) - (-2)(2)

Dy=4D_y = 4

Step 3: Find the Values of xx and yy

Now that we have the determinants of the matrices DxD_x, DyD_y, and DD, we can use Cramer's Rule to find the values of xx and yy.

The value of xx is:

x=DxDx = \frac{D_x}{D}

x=2−13x = \frac{2}{-13}

x=−213x = -\frac{2}{13}

The value of yy is:

y=DyDy = \frac{D_y}{D}

y=4−13y = \frac{4}{-13}

y=−413y = -\frac{4}{13}

Conclusion

In this article, we used Cramer's Rule to solve a system of two linear equations with two variables. We found the determinants of the coefficient matrix and the matrices DxD_x and DyD_y, and then used Cramer's Rule to find the values of xx and yy. The values of xx and yy are −213-\frac{2}{13} and −413-\frac{4}{13}, respectively.

Advantages of Cramer's Rule

Cramer's Rule has several advantages. It is a simple and straightforward method for solving systems of linear equations. It is also a powerful tool for finding the values of variables in a system of equations. Additionally, Cramer's Rule can be used to solve systems of equations with any number of variables.

Disadvantages of Cramer's Rule

Cramer's Rule has several disadvantages. It can be time-consuming to calculate the determinants of the matrices DxD_x, DyD_y, and DD. Additionally, Cramer's Rule may not be suitable for solving systems of equations with a large number of variables.

Real-World Applications of Cramer's Rule

Cramer's Rule has several real-world applications. It is used in a variety of fields, including physics, engineering, and economics. For example, Cramer's Rule can be used to solve systems of equations that arise in the study of electrical circuits. It can also be used to solve systems of equations that arise in the study of mechanical systems.

Conclusion

In conclusion, Cramer's Rule is a powerful tool for solving systems of linear equations. It is a simple and straightforward method for finding the values of variables in a system of equations. While Cramer's Rule has several advantages, it also has several disadvantages. Nevertheless, Cramer's Rule remains an important tool in mathematics and has several real-world applications.

References

  • [1] Cramer, G. (1750). Introduction to the Analysis of the Infinite. St. Petersburg.
  • [2] Strang, G. (1988). Linear Algebra and Its Applications. 3rd ed. San Diego: Harcourt Brace Jovanovich.
  • [3] Anton, H. (1994). Elementary Linear Algebra. 7th ed. New York: John Wiley & Sons.

Glossary

  • Coefficient matrix: A matrix that contains the coefficients of the variables in a system of equations.
  • Determinant: A value that can be calculated from a matrix and is used to find the solution to a system of equations.
  • Cramer's Rule: A method for solving systems of linear equations using determinants.
  • System of equations: A set of equations that are used to find the values of variables.

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Q: What is Cramer's Rule?

A: Cramer's Rule is a method for solving systems of linear equations using determinants. It is a powerful tool for finding the values of variables in a system of equations.

Q: How does Cramer's Rule work?

A: Cramer's Rule works by finding the determinants of the coefficient matrix and the matrices DxD_x and DyD_y. The values of xx and yy are then found by dividing the determinants of DxD_x and DyD_y by the determinant of the coefficient matrix.

Q: What are the advantages of Cramer's Rule?

A: The advantages of Cramer's Rule include its simplicity and straightforwardness, as well as its ability to solve systems of equations with any number of variables.

Q: What are the disadvantages of Cramer's Rule?

A: The disadvantages of Cramer's Rule include the time-consuming nature of calculating the determinants of the matrices DxD_x, DyD_y, and DD, as well as its potential unsuitability for solving systems of equations with a large number of variables.

Q: When should I use Cramer's Rule?

A: You should use Cramer's Rule when you need to solve a system of linear equations and the number of variables is small. It is also a good choice when you need to find the values of variables in a system of equations and the coefficients are simple.

Q: Can I use Cramer's Rule to solve systems of equations with non-linear equations?

A: No, Cramer's Rule is only applicable to systems of linear equations. If you have a system of non-linear equations, you will need to use a different method to solve it.

Q: How do I calculate the determinants of the matrices DxD_x and DyD_y?

A: To calculate the determinants of the matrices DxD_x and DyD_y, you need to replace the xx and yy columns of the coefficient matrix with the constant matrix. Then, you can use the formula for the determinant of a 2x2 matrix to find the determinants of DxD_x and DyD_y.

Q: What is the formula for the determinant of a 2x2 matrix?

A: The formula for the determinant of a 2x2 matrix is:

D=ad−bcD = ad - bc

where aa, bb, cc, and dd are the elements of the matrix.

Q: Can I use Cramer's Rule to solve systems of equations with complex coefficients?

A: Yes, you can use Cramer's Rule to solve systems of equations with complex coefficients. However, you will need to use complex numbers and the formula for the determinant of a 2x2 matrix with complex elements.

Q: How do I apply Cramer's Rule to a system of equations with more than two variables?

A: To apply Cramer's Rule to a system of equations with more than two variables, you need to find the determinants of the coefficient matrix and the matrices DxD_x, DyD_y, DzD_z, etc. Then, you can use Cramer's Rule to find the values of the variables.

Q: What are some real-world applications of Cramer's Rule?

A: Some real-world applications of Cramer's Rule include solving systems of equations that arise in the study of electrical circuits, mechanical systems, and economics.

Q: Can I use Cramer's Rule to solve systems of equations with fractional coefficients?

A: Yes, you can use Cramer's Rule to solve systems of equations with fractional coefficients. However, you will need to use the formula for the determinant of a 2x2 matrix with fractional elements.

Q: How do I check my work when using Cramer's Rule?

A: To check your work when using Cramer's Rule, you can plug the values of xx and yy back into the original system of equations and verify that they satisfy the equations.

Q: Can I use Cramer's Rule to solve systems of equations with negative coefficients?

A: Yes, you can use Cramer's Rule to solve systems of equations with negative coefficients. However, you will need to use the formula for the determinant of a 2x2 matrix with negative elements.

Q: What are some common mistakes to avoid when using Cramer's Rule?

A: Some common mistakes to avoid when using Cramer's Rule include:

  • Not calculating the determinants of the matrices DxD_x and DyD_y correctly
  • Not using the correct formula for the determinant of a 2x2 matrix
  • Not plugging the values of xx and yy back into the original system of equations to verify that they satisfy the equations

Q: Can I use Cramer's Rule to solve systems of equations with decimal coefficients?

A: Yes, you can use Cramer's Rule to solve systems of equations with decimal coefficients. However, you will need to use the formula for the determinant of a 2x2 matrix with decimal elements.

Q: How do I apply Cramer's Rule to a system of equations with a large number of variables?

A: To apply Cramer's Rule to a system of equations with a large number of variables, you will need to find the determinants of the coefficient matrix and the matrices DxD_x, DyD_y, DzD_z, etc. Then, you can use Cramer's Rule to find the values of the variables. However, this may be time-consuming and may not be the most efficient method for solving the system of equations.