Please help!
x-2y+z=7
2x+y-z=0
3x+2y-2z=-2
a. Solve the above system of equations using Gaussian Elimination or Gauss-Jordan Elimination. You must show row operations.
b. Solve the above system of equations using Cramer's Rule.
x=1,y=2,z=3
Sure! Let's start by solving the system of equations using Gaussian Elimination. Here's how you can do it:
a. Gaussian Elimination:
Step 1: Write the augmented matrix for the system of equations:
```
[1 -2 1 | 7]
[2 1 -1 | 0]
[3 2 -2 | -2]
```
Step 2: Perform row operations to get zeros below the pivot in the first column. We can start by making the pivot in the first row, first column equal to 1:
R2 = R2 - 2R1
R3 = R3 - 3R1
```
[1 -2 1 | 7]
[0 5 -3 | -14]
[0 8 -5 | -23]
```
Step 3: Perform row operations to get zeros below the pivot in the second column:
R3 = R3 - (8/5)R2
```
[1 -2 1 | 7]
[0 5 -3 | -14]
[0 0 1 | 1]
```
Step 4: Back-substitute to find the values of x, y, and z:
From the third row, we can see that z = 1.
Substituting this value back into the second row, we can find y:
5y - 3z = -14
5y - 3(1) = -14
5y - 3 = -14
5y = -11
y = -11/5
Substituting the values of y and z into the first row, we can find x:
x - 2y + z = 7
x - 2(-11/5) + 1 = 7
x + 22/5 + 1 = 7
x + 27/5 = 7
x = 7 - 27/5
x = 8/5
So, the solution to the system of equations is x = 8/5, y = -11/5, and z = 1.
b. Cramer's Rule:
To solve the system of equations using Cramer's Rule, we need to find the determinants of matrices obtained by replacing each column in the coefficient matrix with the constant vector.
Step 1: Find the determinant of the coefficient matrix (D):
D = |1 -2 1|
|2 1 -1|
|3 2 -2|
D = 1(1)(-2) + 2(1)(3) + (-1)(2)(2) - 1(-2)(3) - 2(1)(2) - (-1)(2)(1)
D = -2 + 6 - 4 + 6 - 4 + 2
D = 4
Step 2: Find the determinant of the matrix obtained by replacing the first column of the coefficient matrix with the constant vector (Dx):
Dx = |7 -2 1|
|0 1 -1|
|-2 2 -2|
Dx = 7(1)(-2) + 0(1)(-2) + (-2)(2)(-2) - (-2)(1)(-2) - 0(2)(-2) - 7(2)(-1)
Dx = -14 + 8 - 8 + 4 + 0 + 14
Dx = 4
Step 3: Find the determinant of the matrix obtained by replacing the second column of the coefficient matrix with the constant vector (Dy):
Dy = |1 7 1|
|2 0 -1|
|3 -2 -2|
Dy = 1(0)(-2) + 2(-2)(-2) + 3(7)(-1) - 1(-2)(-1) - 2(0)(-1) - 3(-2)(1)
Dy = 0 + 8 - 21 + 2 + 0 + 6
Dy = -5
Step 4: Find the determinant of the matrix obtained by replacing the third column of the coefficient matrix with the constant vector (Dz):
Dz = |1 -2 7|
|2 1 0|
|3 2 -2|
Dz = 1(1)(-2) + 2(3)(-2) + 3(2)(1) - 7(-2)(3) - 2(1)(3) - 1(2)(-2)
Dz = -2 - 12 + 6 + 42 - 6 + 4
Dz = 32
Step 5: Calculate the values of x, y, and z:
x = Dx / D
= 4 / 4
= 1
y = Dy / D
= (-5) / 4
= -5/4
z = Dz / D
= 32 / 4
= 8
So, the solution to the system of equations using Cramer's Rule is x = 1, y = -5/4, and z = 8.
I hope this helps! Let me know if you have any further questions.