Use Cramer's Rule To Solve The System Of Equations.${ \begin{aligned} -4x - 4y + 4z &= 1 \ 5x + 7y - Z &= 2 \ 4x + 4y - 4z &= -1 \end{aligned} }$
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Introduction
Cramer's rule is a method used to solve systems of linear equations. It involves calculating the determinants of a matrix and its modified versions to find the values of the variables. In this article, we will use Cramer's rule to solve the system of equations:
\begin{aligned} -4x - 4y + 4z &= 1 \ 5x + 7y - z &= 2 \ 4x + 4y - 4z &= -1 \end{aligned} }
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 and involves calculating the determinants of a matrix and its modified versions. The rule states that if we have a system of linear equations in the form:
{ \begin{aligned} a_1x + b_1y + c_1z &= d_1 \\ a_2x + b_2y + c_2z &= d_2 \\ a_3x + b_3y + c_3z &= d_3 \end{aligned} \}$ $ Then the values of x, y, and z can be found using the following formulas: ${ \begin{aligned} x &= \frac{D_x}{D} \\ y &= \frac{D_y}{D} \\ z &= \frac{D_z}{D} \end{aligned} \}$ $ where D is the determinant of the coefficient matrix, and Dx, Dy, and Dz are the determinants of the modified matrices obtained by replacing the first, second, and third columns of the coefficient matrix with the constant terms. ## **Calculating the Determinant of the Coefficient Matrix** --------------------------------------------------- The coefficient matrix of the given system of equations is: ${ \begin{aligned} \begin{bmatrix} -4 & -4 & 4 \\ 5 & 7 & -1 \\ 4 & 4 & -4 \end{bmatrix} \end{aligned} \}$ $ To calculate the determinant of this matrix, we can use the formula for the determinant of a 3x3 matrix: ${ \begin{aligned} D &= a(ei - fh) - b(di - fg) + c(dh - eg) \\ &= -4(7(-4) - (-1)4) - (-4)(5(-4) - (-1)4) + 4(5(4) - 7(4)) \\ &= -4(-28 + 4) + 4(-20 + 4) + 4(20 - 28) \\ &= -4(-24) + 4(-16) + 4(-8) \\ &= 96 - 64 - 32 \\ &= 0 \end{aligned} \}$ $ ## **Calculating the Determinants of the Modified Matrices** --------------------------------------------------- To calculate the determinants of the modified matrices, we need to replace the first, second, and third columns of the coefficient matrix with the constant terms. ### **Determinant of the Modified Matrix for x** The modified matrix for x is obtained by replacing the first column of the coefficient matrix with the constant terms: ${ \begin{aligned} \begin{bmatrix} 1 & -4 & 4 \\ 2 & 7 & -1 \\ -1 & 4 & -4 \end{bmatrix} \end{aligned} \}$ $ To calculate the determinant of this matrix, we can use the formula for the determinant of a 3x3 matrix: ${ \begin{aligned} D_x &= 1(7(-4) - (-1)4) - (-4)(2(-4) - (-1)(-1)) + 4(2(4) - 7(-1)) \\ &= 1(-28 + 4) + 4(-8 - 1) + 4(8 + 7) \\ &= 1(-24) + 4(-9) + 4(15) \\ &= -24 - 36 + 60 \\ &= 0 \end{aligned} \}$ $ ### **Determinant of the Modified Matrix for y** The modified matrix for y is obtained by replacing the second column of the coefficient matrix with the constant terms: ${ \begin{aligned} \begin{bmatrix} -4 & 1 & 4 \\ 5 & 2 & -1 \\ 4 & -1 & -4 \end{bmatrix} \end{aligned} \}$ $ To calculate the determinant of this matrix, we can use the formula for the determinant of a 3x3 matrix: ${ \begin{aligned} D_y &= -4(2(-4) - (-1)(-1)) - 1(5(-4) - (-1)4) + 4(5(-1) - 2(4)) \\ &= -4(-8 - 1) - 1(-20 - 4) + 4(-5 - 8) \\ &= -4(-9) - 1(-24) + 4(-13) \\ &= 36 + 24 - 52 \\ &= 8 \end{aligned} \}$ $ ### **Determinant of the Modified Matrix for z** The modified matrix for z is obtained by replacing the third column of the coefficient matrix with the constant terms: ${ \begin{aligned} \begin{bmatrix} -4 & -4 & 1 \\ 5 & 7 & 2 \\ 4 & 4 & -1 \end{bmatrix} \end{aligned} \}$ $ To calculate the determinant of this matrix, we can use the formula for the determinant of a 3x3 matrix: ${ \begin{aligned} D_z &= -4(7(-1) - 2(4)) - (-4)(5(-1) - 2(4)) + 1(5(4) - 7(4)) \\ &= -4(-7 - 8) + 4(-5 - 8) + 1(20 - 28) \\ &= -4(-15) + 4(-13) + 1(-8) \\ &= 60 - 52 - 8 \\ &= 0 \end{aligned} \}$ $ ## **Finding the Values of x, y, and z** -------------------------------------- Now that we have calculated the determinants of the coefficient matrix and the modified matrices, we can use Cramer's rule to find the values of x, y, and z. ${ \begin{aligned} x &= \frac{D_x}{D} = \frac{0}{0} \\ y &= \frac{D_y}{D} = \frac{8}{0} \\ z &= \frac{D_z}{D} = \frac{0}{0} \end{aligned} \}$ $ Since the determinants of the coefficient matrix and the modified matrices are all zero, the system of equations has no solution. ## **Conclusion** ---------- In this article, we used Cramer's rule to solve the system of equations: ${ \begin{aligned} -4x - 4y + 4z &= 1 \\ 5x + 7y - z &= 2 \\ 4x + 4y - 4z &= -1 \end{aligned} \}$ $ However, since the determinants of the coefficient matrix and the modified matrices are all zero, the system of equations has no solution. This highlights the importance of checking the determinants before applying Cramer's rule to solve a system of equations. ## **References** -------------- * Cramer, G. (1750). _Introduction à l'analyse des lignes courbes algébriques_. Paris: Imprimerie Royale. * Strang, G. (1988). _Linear Algebra and Its Applications_. 3rd ed. San Diego: Harcourt Brace Jovanovich. * Anton, H. (1994). _Elementary Linear Algebra_. 7th ed. New York: Wiley.<br/> # **Frequently Asked Questions About Cramer's Rule** ===================================================== ## **Q: What is Cramer's Rule?** --------------------------- A: Cramer's rule is a method used to solve systems of linear equations. It involves calculating the determinants of a matrix and its modified versions to find the values of the variables. ## **Q: When can I use Cramer's Rule?** ----------------------------------- A: You can use Cramer's rule to solve systems of linear equations that have a unique solution. However, if the system has no solution or an infinite number of solutions, Cramer's rule will not work. ## **Q: How do I calculate the determinant of a matrix using Cramer's Rule?** ------------------------------------------------------------------- A: To calculate the determinant of a matrix using Cramer's rule, you need to follow these steps: 1. Write down the matrix. 2. Replace each column of the matrix with the constant terms of the system of equations. 3. Calculate the determinant of the modified matrix. 4. Divide the determinant of the modified matrix by the determinant of the original matrix. ## **Q: What is the formula for Cramer's Rule?** ----------------------------------------- A: The formula for Cramer's rule is: ${ \begin{aligned} x &= \frac{D_x}{D} \\ y &= \frac{D_y}{D} \\ z &= \frac{D_z}{D} \end{aligned} \}$ $ where D is the determinant of the coefficient matrix, and Dx, Dy, and Dz are the determinants of the modified matrices obtained by replacing the first, second, and third columns of the coefficient matrix with the constant terms. ## **Q: What happens if the determinant of the coefficient matrix is zero?** ------------------------------------------------------------------- A: If the determinant of the coefficient matrix is zero, then the system of equations has no solution or an infinite number of solutions. In this case, Cramer's rule will not work. ## **Q: Can I use Cramer's Rule to solve systems of equations with more than three variables?** ----------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with more than three variables. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a reliable method for solving systems of equations?** -------------------------------------------------------------------------------- A: Cramer's rule is a reliable method for solving systems of equations, but it has its limitations. If the system of equations has no solution or an infinite number of solutions, Cramer's rule will not work. Additionally, Cramer's rule can be computationally intensive for large systems of equations. ## **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, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a good method for solving systems of equations in real-world applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule can be a good method for solving systems of equations in real-world applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **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 other methods such as numerical methods or algebraic methods to solve the system. ## **Q: Is Cramer's Rule a difficult method to learn?** --------------------------------------------------- A: Cramer's rule is a relatively simple method to learn, but it does require a good understanding of linear algebra and matrix operations. If you have a good grasp of these concepts, you should be able to learn Cramer's rule quickly. ## **Q: Can I use Cramer's Rule to solve systems of equations with multiple variables?** ----------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with multiple variables. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a good method for solving systems of equations in engineering applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule can be a good method for solving systems of equations in engineering applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with integer coefficients?** ----------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with integer coefficients. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a reliable method for solving systems of equations in scientific applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule is a reliable method for solving systems of equations in scientific applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with rational coefficients?** ----------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with rational coefficients. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a good method for solving systems of equations in computer science applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule can be a good method for solving systems of equations in computer science applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with complex coefficients and rational coefficients?** ----------------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with complex coefficients and rational coefficients. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a reliable method for solving systems of equations in economics applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule is a reliable method for solving systems of equations in economics applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with non-square matrices?** ----------------------------------------------------------------------------------- A: No, Cramer's rule is only applicable to square matrices. If you have a non-square matrix, you will need to use other methods such as Gaussian elimination or LU decomposition to solve the system. ## **Q: Is Cramer's Rule a good method for solving systems of equations in physics applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule can be a good method for solving systems of equations in physics applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with matrices with complex entries?** ----------------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with matrices with complex entries. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a reliable method for solving systems of equations in engineering applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule is a reliable method for solving systems of equations in engineering applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with matrices with rational entries?** ----------------------------------------------------------------------------------------- A: Yes, you can use Cramer's rule to solve systems of equations with matrices with rational entries. However, the process becomes more complicated and requires more calculations. ## **Q: Is Cramer's Rule a good method for solving systems of equations in computer science applications?** ----------------------------------------------------------------------------------------- A: Cramer's rule can be a good method for solving systems of equations in computer science applications, but it depends on the specific problem. If the system of equations is small and has a unique solution, Cramer's rule can be a good choice. However, if the system of equations is large or has no solution or an infinite number of solutions, other methods such as Gaussian elimination or LU decomposition may be more suitable. ## **Q: Can I use Cramer's Rule to solve systems of equations with matrices with complex entries and rational entries?</span></p>