(1 pt) Consider the following system \(2x + 2y - z = 1\ \(3x + 7y + 8z = 1\ \(5x + 5y = -2\ and let \(A\) be its coefficient matrix. (a) \(det(A) = \) (b) Solve for \(x\) using Cramer's Rule. \(det(A_1) = \) and \(x = \) (c) Solve for \(y\) using Cramer's Rule. \(det(A_2) = \) and \(y = \) (d) Solve for \(z\) using Cramer's Rule. \(det(A_3) = \) and \(z = \)
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Step 1: The coefficient matrix A is: A = [[2, 2, -1], [3, 7, 8], [5, 5, 0]] Show more…
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Consider the system $$\left\{\begin{aligned} x+2 y+6 z &=5 \\ -3 x-6 y+5 z &=8 \\ 2 x+6 y+9 z &=7 \end{aligned}\right.$$ a. Verify that $x=-1, y=0, z=1$ is a solution of the system. b. Find the determinant of the coefficient matrix. c. Without solving the system, determine whether there are any other tions. d. Can Cramer's Rule be used to solve this system? Why or why not?
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The determinant of the coefficient matrix is |A| = | -1 2 -3 2 0 1 3 -4 4 | = 10. Since |A| ≠ 0, you know that the system has a unique solution, and Cramer’s Rule can be applied to solve it as follows: x_1 = | 1 2 -3 0 0 1 2 -4 4 | / 10 = 4/5, x_2 = | -1 1 -3 2 0 1 3 2 4 | / 10 = -3/2, x_3 = | -1 2 1 2 0 0 3 -4 2 | / 10 = -8/5. (a) Use Cramer’s Rule to solve the following systems of linear equations, if possible. i. 4x_1 - 2x_2 = 10 3x_1 - 5x_2 = 11 ii. 3x_1 + 3x_2 + 5x_3 = 1 3x_1 + 5x_2 + 9x_3 = 2 5x_1 + 9x_2 + 17x_3 = 4 iii. 2x_1 + 3x_2 + 5x_3 = 4 3x_1 + 5x_2 + 9x_3 = 7 5x_1 + 9x_2 + 17x_3 = 13 (b) ⋆ Explain why Cramer’s Rule works. [Hint: Use the problem 8.]
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