For what values of (¢ belongs to R) the following system of equations has (1)no solution (2)a unique solution (3)infinitely many solutions?
(5-¢)x+4y+2z=4
4x+(5-¢)y+2z=4
2x+2y+(2-¢)z=2
To determine the values of ¢ for which the given system of equations has:
(1) No solution
(2) A unique solution
(3) Infinitely many solutions
We can start by rewriting the system of equations in matrix form:
| 5-¢ 4 2 | | x | | 4 |
| 4 5-¢ 2 | × | y | = | 4 |
| 2 2 2-¢| | z | | 2 |
Now, we can use the theory of augmented matrices and perform Gaussian elimination to determine the possibilities.
Step 1: Set up the augmented matrix
[ 5-¢ 4 2 | 4 ]
[ 4 5-¢ 2 | 4 ]
[ 2 2 2-¢| 2 ]
Step 2: Perform row operations to put the matrix in reduced-row echelon form
We'll use elementary row operations to manipulate the matrix into the reduced-row echelon form (RREF).
Step 3: Analyze the matrix
a) If the RREF of the matrix has any row with all zeros except on the right side (i.e., inconsistent row), the system has no solution.
b) If there are no inconsistent rows, count the number of leading 1's (pivot positions) in the RREF matrix:
- If the number of leading 1's is equal to the number of variables (3 in this case), then the system has a unique solution.
- If the number of leading 1's is less than the number of variables, the system has infinitely many solutions.
For each case:
(1) No solution:
If the RREF matrix has an inconsistent row, then the system has no solution. In this case, there will be at least one row in the RREF matrix with all zeros except on the right side. This occurs when the coefficients of x, y, and z combine to form an inconsistent equation, meaning the equations have contradictory conditions.
(2) A unique solution:
If the number of leading 1's in the RREF matrix is equal to the number of variables (3), then the system has a unique solution. This means the system can be solved for x, y, and z, resulting in a unique set of values for the variables.
(3) Infinitely many solutions:
If the number of leading 1's in the RREF matrix is less than the number of variables (3), then the system has infinitely many solutions. In this case, there will be a free variable that can take on any value, resulting in infinitely many possible solutions.
By following the steps outlined above and analyzing the obtained RREF matrix, you can determine the values of ¢ that correspond to each case and find the desired solutions to the system of equations.