Question

Apply crammer's rules for linear equations x+y-z=1 8x+3y-6z=1 -4x-y+3z=1

Answer 1

To apply Cramer's Rule to solve the system of equations, write the equations in matrix form, calculate the determinants, and use the determinants to find the solution for the variables x, y, and z.

Step-by-step explanation:

To apply Cramer's Rule to solve the system of equations:

1. Write the system of equations in the form AX = B, where A is the matrix of coefficients, X is the column matrix of variables, and B is the column matrix of constants.
2. Calculate the determinant of the coefficient matrix A.
3. Replace each column of the coefficient matrix A with the column matrix B, and calculate the determinant of each resulting matrix.
4. Divide each determinant calculated in step 3 by the determinant of A calculated in step 2.
5. The resulting values are the solutions for the variables x, y, and z.

For the given system of equations:

x + y - z = 1

8x + 3y - 6z = 1

-4x - y + 3z = 1

The coefficient matrix A is:

[1 1 -1]
[8 3 -6]
[-4 -1 3]

The matrix of constants B is:

[1]
[1]
[1]

The determinant of A is: 34

Using Cramer's Rule, we can calculate the determinants of the matrices obtained by replacing the columns of A with B:

dx = [1 1 -1]
[1 3 -6]
[1 -1 3]

dy = [1 1 -1]
[8 1 -6]
[-4 1 3]

dz = [1 1 -1]
[8 3 1]
[-4 -1 1]

Dividing each determinant by the determinant of A, we get:

x = dx / determinant_A = 1 / 34 = 0.0294

y = dy / determinant_A = 5 / 34 = 0.1471

z = dz / determinant_A = 4 / 34 = 0.1176