Solve Using a Matrix with Cramer's Rule 5x+2y-z=1 , x-2y+2z=0 , 2x-3y+z=-2

Problem

{[5*x+2*y−z=1],[x−2*y+2*z=0],[2*x−3*y+z=−2])

Solution

1. Identify the coefficient matrix A and the constant vector B from the system of equations.

A=[[5,2,−1],[1,−2,2],[2,−3,1]]

B=[[1],[0],[−2]]

2. Calculate the determinant of the coefficient matrix D=det(A) using expansion by the first row.

D=5*((−2)*(1)−(2)*(−3))−2*((1)*(1)−(2)*(2))+(−1)*((1)*(−3)−(−2)*(2))

D=5*(−2+6)−2*(1−4)−1*(−3+4)

D=5*(4)−2*(−3)−1*(1)

D=20+6−1=25

3. Calculate the determinant (D_x) by replacing the first column of A with the constant vector B

(D_x)=[[1,2,−1],[0,−2,2],[−2,−3,1]]

(D_x)=1*((−2)*(1)−(2)*(−3))−2*((0)*(1)−(2)*(−2))+(−1)*((0)*(−3)−(−2)*(−2))

(D_x)=1*(4)−2*(4)−1*(−4)

(D_x)=4−8+4=0

4. Calculate the determinant (D_y) by replacing the second column of A with the constant vector B

(D_y)=[[5,1,−1],[1,0,2],[2,−2,1]]

(D_y)=5*((0)*(1)−(2)*(−2))−1*((1)*(1)−(2)*(2))+(−1)*((1)*(−2)−(0)*(2))

(D_y)=5*(4)−1*(−3)−1*(−2)

(D_y)=20+3+2=25

5. Calculate the determinant (D_z) by replacing the third column of A with the constant vector B

(D_z)=[[5,2,1],[1,−2,0],[2,−3,−2]]

(D_z)=5*((−2)*(−2)−(0)*(−3))−2*((1)*(−2)−(0)*(2))+1*((1)*(−3)−(−2)*(2))

(D_z)=5*(4)−2*(−2)+1*(1)

(D_z)=20+4+1=25

6. Apply Cramer's Rule to find the values of x y and z by dividing the specific determinants by D

x=(D_x)/D=0/25=0

y=(D_y)/D=25/25=1

z=(D_z)/D=25/25=1

Final Answer

(x,y,z)=(0,1,1)

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