The solution to the system of equations is x = -4.5, y = 4, and z = 0.5.
The solution to the system of equations using matrices, along with an explanation of the steps involved:
Step 1: Represent the system of equations as a matrix equation
The given system of equations can be represented as a matrix equation in the form:
AX = B
where:
A is the coefficient matrix, which consists of the coefficients of the variables in the equations
X is the column vector of variables, which consists of the variables to be solved for
B is the column vector of right-hand side constants
In this case, the coefficient matrix A, the column vector of variables X, and the column vector of right-hand side constants B are given as follows:
A = [[2, -1, 2],
[-1, 1, -1],
[-1, 0, -3]]
X = [[x],
[y],
[z]]
B = [[6],
[-1],
[-6]]
Therefore, the matrix equation representing the system of equations is:
[[2, -1, 2],
[-1, 1, -1],
[-1, 0, -3]] * [[x],
[y],
[z]] = [[6],
[-1],
[-6]]
Step 2: Solve the matrix equation using matrix inversion
To solve the system of equations using matrices, we can invert the coefficient matrix A and multiply both sides of the matrix equation by the inverse of A. This gives us:
X = A^(-1) * B
where A^(-1) is the inverse of the coefficient matrix A.
In this case, the inverse of the coefficient matrix A can be calculated using a matrix inversion algorithm, such as the Gaussian elimination method. The inverse of A is:
A^(-1) = [[0.5, 0.5, -0.5],
[-0.5, 1, -0.5],
[0.5, 0, -0.25]]
Multiplying both sides of the matrix equation by A^(-1), we get:
[[x],
[y],
[z]] = [[0.5, 0.5, -0.5],
[-0.5, 1, -0.5],
[0.5, 0, -0.25]] * [[6],
[-1],
[-6]]
Expanding the matrix multiplication, we get the following system of equations:
0.5x + 0.5y - 0.5z = 6
-0.5x + y - 0.5z = -1
0.5x - 0.25z = -6
Step 3: Solve the system of equations using back-substitution
We can now solve the system of equations using back-substitution. To do this, we first solve the third equation for x:
x = -4.5
Substituting this value of x into the second equation, we get
4.5y - 0.5z = -1
Solving for y, we get:
y = 4
Substituting the values of x and y into the first equation, we get:
-9 - 4z = 6
Solving for z, we get:
z = 0.5
Therefore, the solution to the system of equations is x = -4.5, y = 4, and z = 0.5.