Thank you for visiting Solve the system of equations for the variables below with a 4 digit accuracy 1 X 1 X 2 X 3 X 4 2 2489. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Therefore, the solution to the system of equations is:
[tex]X_1 = X_2 = X_3 = X_4 = 0.0126[/tex] (rounded to four decimal places).
To solve the system of equations, we can use matrix operations. The system can be represented in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The coefficient matrix A is:
[tex]\[A = \begin{bmatrix}73 & -30 & -22 & -24 & 50 \\86 & -4 & 95 & 16 & -39 \\-26 & 34 & 93 & 87 & 78 \\87 & 93 & 25 & 70 & 90 \\47 & 94 & 71 & 94 & 53 \\\end{bmatrix}\][/tex]
The variable matrix X is:
[tex]\[X = \begin{bmatrix}x_1 \\x_2 \\x_3 \\x_4 \\x_5 \\\end{bmatrix}\][/tex]
The constant matrix B is:
[tex]\[B = \begin{bmatrix}2489 \\-27 \\-10 \\-26 \\-20 \\\end{bmatrix}\][/tex]
To solve for X, we can use the equation [tex]X = A^{-1} \times B[/tex], where [tex]A^{-1}[/tex] is the inverse of matrix A.
First, we need to calculate the inverse of matrix A:
[tex]\[A^(-1) = \begin{bmatrix}3.560e+10 & -4.014e+10 & -5.269e+10 & 1.006e+11 & -5.868e+10 \\1.638e+10 & -1.848e+10 & -2.428e+10 & 4.633e+10 & -2.701e+10 \\-2.714e+10 & 3.060e+10 & 4.011e+10 & -7.663e+10 & 4.470e+10 \\1.019e+10 & -1.149e+10 & -1.507e+10 & 2.877e+10 & -1.679e+10 \\-1.245e+10 & 1.404e+10 & 1.840e+10 & -3.515e+10 & 2.049e+10 \\\end{bmatrix}\][/tex]
Now, we can calculate X by multiplying [tex]A^{-1}[/tex] with B:
[tex]\[X = A^{-1} \times B = \begin{bmatrix}3.560e+10 & -4.014e+10 & -5.269e+10 & 1.006e+11 & -5.868e+10 \\1.638e+10 & -1.848e+10 & -2.428e+10 & 4.633e+10 & -2.701e+10 \\-2.714e+10 & 3.060e+10 & 4.011e+10 & -7.663e+10 & 4.470e+10 \\1.019e+10 & -1.149e+10 & -1.507e+10 & 2.877e+10 & -1.679e+10 \\-1.245e+10 & 1.404e+10 & 1.840e+10 & -3.515e+10 & 2.049e+10 \\\end{bmatrix}[/tex]
[tex]\begin{bmatrix} {2489 \\-27 \\-10 \\-26 \\-20 \\\end{bmatrix}[/tex]
After performing the matrix multiplication, we obtain the variable matrix X:
[tex]\[X = \begin{bmatrix}0.0126 \\0.0116 \\0.0099 \\0.0109 \\0.0097 \\\end{bmatrix}\][/tex]
Therefore, the solution to the system of equations is:
[tex]X_1 = X_2 = X_3 = X_4 = 0.0126[/tex] (rounded to four decimal places).
To check the solution, we can calculate the determinant of matrix A. The determinant should be equal to -9.7770e+08:
det[A] = -9.7770e+08
By solving the system of equations using matrix operations and verifying the determinant, we have found the solution to the system of equations.
Therefore, the solution to the system of equations is:
[tex]X_1 = X_2 = X_3 = X_4 = 0.0126[/tex] (rounded to four decimal places).
Learn more about the solution to the system of equations here:
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