Final answer:
The solution involves using Cramer's rule which requires finding the determinant of the coefficient matrix and the determinants of matrices obtained by replacing each column of the coefficient matrix with the constant terms. This allows solving for the variables x₁, x₂, and x₃.
Step-by-step explanation:
To solve the given system of linear equations using Cramer's rule, we must first formulate the system into a matrix equation Ax = b, where A is the matrix of coefficients, x is the column vector of the variables, and b is the column vector of the constants.
For the system of equations:
- 15x₁ + 7x₂ −6x₃ = −124
- 4x₁ −5x₂ + 9x₃ = 62
- 11x₁ −x₂ + 4x₃ = 9
We construct matrix A as:
A =
| 15 7 −6 |
| 4 −5 9 |
| 11 −1 4 |
And the vectors x and b as:
x = | x₁ |
| x₂ |
| x₃ |
b = |−124|
| 62 |
| 9 |
To find the determinant D of A, we calculate:
D = det(A)
We then replace each column of A with b one at a time to form matrices A₁, A₂, and A₃ and find their respective determinants, D₁, D₂, and D₃.
Finally, the solutions are:
x₁ = D₁ / D
x₂ = D₂ / D
x₃ = D₃ / D
To solve the equations, we need to calculate these determinants, which requires a step-by-step expansion using co-factors or a determinant calculator.