Question

Use the Crammer's to compute the solutions of the following systems.
4x - y + 2z
x + 2y - 2z = 0
-x + y + z = 5

Answer 1

you need to divide the x,y,z determinants by the main determinant.

Answer 2

To solve this system of equations using Cramer's rule, we first need to set up the matrix with coefficients of the variables on the left side and the constant terms on the right side. The matrix will look like:

| 4 -1 2 |
| 1 2 -2 |
| -1 1 1 |

Now, we will also calculate the determinants to determine the values of x, y, and z. To calculate the determinant of the main matrix, we get:

| 4 -1 2 |
| 1 2 -2 |
| -1 1 1 |

= 4(2*1 - (-2)*1) - (-1)(1*1 - (-2)(-1)) + 2(1*1 - 2*1)
= 4(2 + 2) - (-1)(1 + 2) + 2(1 - 2)
= 16 - 3 + 2
= 15

Next, we calculate the determinants for x, y, and z by replacing the first column of the main matrix with the constants matrix (after the equal sign). For x, we get:

| 0 -1 2 |
| 2 2 -2 |
| 5 1 1 |

= 0(2*1 - (-2)*1) - (-1)(2*1 - (-2)*5) + 2(2*1 - 5*1)
= 0 - (-4) + 2
= 6

For y, we get:

| 4 0 2 |
| 1 5 -2 |
| -1 5 1 |

= 4(5*1 - (-2)*5) - 0(1*1 - (-2)*(-1)) + 2(1*5 - 5*1)
= 4(5 + 10) + 2
= 60

And for z, we get:

| 4 -1 0 |
| 1 2 5 |
| -1 1 5 |

= 4(2*5 - 5*1) - (-1)(1*5 - 5*0) + 0(1*1 - 2*0)
= 20 - 5 + 0
= 15

Therefore, the solutions to this system of equations are x = 6, y = 60, and z = 15.

Answer 3

Apologies for the oversight. Let's divide the determinants by the main determinant to get the correct values of x, y, and z.

x = determinant(x) / determinant(main) = 6 / 15 = 2/5
y = determinant(y) / determinant(main) = 60 / 15 = 4
z = determinant(z) / determinant(main) = 15 / 15 = 1

Therefore, the solutions to the system of equations are x = 2/5, y = 4, and z = 1.