caluclas (intersection of three planes) HELP!!!?
in the equations k is a real number
1.x-2y+z=4
2.x-y-z=3
3.x+y+kz=1
for what values of k does the system
1.have exactly one solution?
2.have an in finite number of solution?
To determine the values of k for which the given system of equations has one solution or an infinite number of solutions, we need to analyze the intersection of the three planes represented by the equations.
To solve the system, we'll use the method of row reduction or Gaussian elimination. We'll represent the system of equations as an augmented matrix and perform row operations until we obtain row-echelon form. Then, based on the resulting matrix, we'll determine the solutions.
Let's start by writing the augmented matrix for the system of equations:
1 -2 1 | 4
1 -1 -1 | 3
1 1 k | 1
Perform elementary row operations to get the augmented matrix in row-echelon form:
R2 = R2 - R1
R3 = R3 - R1
1 -2 1 | 4
0 1 -2 | -1
0 3 k-1 | -3
Next, perform additional row operations to further simplify the matrix:
R3 = R3 - 3R2
1 -2 1 | 4
0 1 -2 | -1
0 0 3k-5 | 0
Now, let's consider the two cases separately:
Case 1: The system has exactly one solution.
For the system to have a unique solution, the row-echelon form should have only a single nonzero entry in each row. As we can see, this condition is satisfied for the third row only if 3k - 5 = 0.
Therefore, for the system to have exactly one solution, k must be such that 3k - 5 = 0. Solving this equation:
3k - 5 = 0
3k = 5
k = 5/3
So, the system has exactly one solution if k = 5/3.
Case 2: The system has an infinite number of solutions.
For the system to have an infinite number of solutions, there should be a row of zeros at some point in the row-echelon form.
In our current row-echelon form, this can only occur if 3k - 5 = 0 and the last row is also all zeros. Since the last row is not all zeros, it means there is no value of k for which the system has an infinite number of solutions.
In summary:
1. The system has exactly one solution when k = 5/3.
2. The system does not have an infinite number of solutions for any value of k.
I hope this explanation helps you understand how to approach and solve this type of problem.