Answer:
(a) (x, y, z) = (5, -1, -1)
(b) singular; the lines of intersection are parallel
Step-by-step explanation:
You want the solutions to the following systems of equations:
A calculator conveniently reduces the augmented coefficient matrices to reduced row-echelon form. When the left part of the result is the identity matrix, the equations have the solution given in the rightmost column.
The solution is (x, y, z) = (5, -1, -1).
When the left part of the result has a bottom row of zero, and the lower right number is 1, it means the system is singular (has a zero determinant), and the equations are inconsistent. The graph of the equations in the second attachment shows the planes intersect in three parallel lines, so there is no point that satisfies all three equations.
To solve the given systems of equations, you can use substitution or elimination methods, but it is crucial to check if the system is singular by ensuring the determinant of the coefficient matrix is non-zero.
To solve the given systems of equations, one approach is to use matrix operations or classical elimination methods. In the first system (a), we have the equations:
x+2y+3z=0
-2x-2y+z=-9
x+4y+9z=-8.
You can use the substitution or elimination method to find the values of x, y, and z. If these equations are singular, it means that they do not have a unique solution, which could be due to the equations being dependent, leading to infinite solutions or inconsistent, leading to no solution.
For the second system (b):
x+2y+3z=0
-2x-2y+z=-9
x+4y+10z=-8,
you would follow a similar process. Notice if the determinant of the coefficient matrix is zero, which would indicate that the system is singular, and it may have no solution or infinitely many solutions.
It is important to remember that for the system to be non-singular, the coefficient matrix must have a non-zero determinant.
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