For the following plane equations:
x + 2y + 3z = 1
2x + 5y + 3z = -1
x + bz = c
Find all values of b and c such that all planes are the same planes.
I've reduced it into the row echelon form but don't know what to do next. Thanks beforehand
"...that all planes are the same planes. "
this implies that all equations must be multiples of each other, but...
the first and second equations have different normals , so they are not parallel
Unless I am missing something here, this question is not possible
To find the values of b and c such that all planes are the same, we can solve the system of equations using the row echelon form. Here's how you can proceed:
1. Write the system of equations in augmented matrix form:
[1 2 3 | 1]
[2 5 3 | -1]
[1 0 b | c]
2. Perform row operations to reduce the matrix to row echelon form. Start by swapping rows to bring the row with the nonzero entry in the leftmost column to the first row.
[1 0 b | c]
[2 5 3 | -1]
[1 2 3 | 1]
3. To eliminate the entries below the leading entry (1) in the first column, subtract twice the first row from the second row, and subtract the first row from the third row.
[1 0 b | c]
[0 5-2b 3-2b | -1-2c]
[0 2 3-b | 1-c]
4. To eliminate the entry below the leading entry (5-2b) in the second row, subtract 2/5 times the second row from the third row.
[1 0 b | c]
[0 5-2b 3-2b | -1-2c]
[0 0 (3-b)-(2/5)(3-2b) | 1-c-(2/5)(-1-2c)]
5. Simplify the last row.
[1 0 b | c]
[0 5-2b 3-2b | -1-2c]
[0 0 3/5b-1 | (7+8c)/5]
6. To ensure that all planes are the same, the last row should be all zeros except for the last entry, which should also be zero. This gives us the equation:
3/5b - 1 = 0, or 3b = 5.
Solving for b, we get b = 5/3.
7. Substituting this value of b into the last entry of the last row equation, we get:
(7 + 8c)/5 = 0, or 7 + 8c = 0.
Solving for c, we get c = -7/8.
Therefore, the values of b and c such that all planes are the same planes are b = 5/3 and c = -7/8.