3. Solv by Cramer's Role:
2x-3y+4z=3
x+4y-5z=0
5x-y+z=5
To solve the given system of equations using Cramer's Rule, we first need to find the determinants of the coefficient matrix and each of the derived matrices.
Step 1: Determinants
Let's define the coefficient matrix A, variable matrix X, and constant matrix B:
A = [[2, -3, 4], [1, 4, -5], [5, -1, 1]]
X = [[x], [y], [z]]
B = [[3], [0], [5]]
Now, let's find the determinant of the coefficient matrix A, denoted as |A|.
|A| = det(A) = 2(4(1) - (-5)(-1)) - (-3)(1(1) - (-5)(5)) + 4(1(-1) - 4(5))
= 2(4 + 5) - (-3)(1 + 25) + 4(-1 - 20)
= 18 - (-78) - 84
= 18 + 78 - 84
= 12
Step 2: Determinants of Derived Matrices
Now, we need to find the determinants of the derived matrices by replacing the columns of A with the constant matrix B.
To find the determinant of the first derived matrix, replace the first column of A with B:
|A1| = det([[3, -3, 4], [0, 4, -5], [5, -1, 1]])
= 3(4(1) - (-5)(-1)) - (-3)(0(1) - (-5)(5)) + 4(0(1) - 4(5))
= 3(4 + 5) - (-3)(0 + 25) + 4(0 - 20)
= 27 - 0 - 80
= -53
To find the determinant of the second derived matrix, replace the second column of A with B:
|A2| = det([[2, 3, 4], [1, 0, -5], [5, 5, 1]])
= 2(0(1) - (-5)(5)) - 3(1(1) - (-5)(5)) + 4(1(5) - 0(5))
= 2(0 + 25) - 3(1 + 25) + 4(5 - 0)
= 50 - 78 + 20
= -8
To find the determinant of the third derived matrix, replace the third column of A with B:
|A3| = det([[2, -3, 3], [1, 4, 0], [5, -1, 5]])
= 2(4(5) - 0(-1)) - (-3)(1(5) - 0(5)) + 3(1(-1) - 4(0))
= 2(20) - (-3)(5) + 3(-1)
= 40 + 15 - 3
= 52
Step 3: Solve for variables
The solutions for the variables x, y, and z are obtained by evaluating the ratios of the determinants of the derived matrices with the determinant of the coefficient matrix:
x = |A1| / |A| = -53 / 12
y = |A2| / |A| = -8 / 12
z = |A3| / |A| = 52 / 12
Simplifying the solutions:
x = -4.42
y = -0.67
z = 4.33
Therefore, the solution to the system of equations is x ≈ -4.42, y ≈ -0.67, and z ≈ 4.33.