For which values of 'a' will the following system have no solutions? Exactly one solution? Infinitely many solutions?
x + 2y - 3z = 4
3x - y +5z = 2
4x + y + (a^2-14)z = a+2
I do not see how the lines can be parallel and the only real problem I see so far is singularity when a = +/-4
To determine the number of solutions for this system, we need to examine the coefficient matrix and augmented matrix.
The coefficient matrix, let's call it A, is:
A = [[1, 2, -3],
[3, -1, 5],
[4, 1, a^2 - 14]]
The augmented matrix, let's call it B, is obtained by appending the column on the right of the coefficient matrix:
B = [[1, 2, -3, 4],
[3, -1, 5, 2],
[4, 1, a^2 - 14, a+2]]
Next, we can perform row operations on the augmented matrix to determine its row-echelon form.
1. Row2 = Row2 - 3 * Row1
B = [[1, 2, -3, 4],
[0, -7, 14, -10],
[4, 1, a^2 - 14, a+2]]
2. Row3 = Row3 - 4 * Row1
B = [[1, 2, -3, 4],
[0, -7, 14, -10],
[0, -7, a^2 + 2, a-6]]
3. Row3 = Row3 - Row2
B = [[1, 2, -3, 4],
[0, -7, 14, -10],
[0, 0, a^2 - 12, a+4]]
At this point, B is in row-echelon form. Now, let's analyze the possibilities for different values of 'a':
1. No Solution:
If a^2 - 12 ≠ 0, then the system will have no solution because the third row has a leading 1 in the fourth column (the augmented column). This means there is a row of zeros (the last row) that doesn't match the augmented column.
2. Exactly One Solution:
If a^2 - 12 = 0 and a + 4 ≠ 0, then the system will have exactly one solution because all the variables (x, y, z) are leading variables and there are no free variables.
3. Infinitely Many Solutions:
If a^2 - 12 = 0 and a + 4 = 0, then the system will have infinitely many solutions because there is at least one free variable (z).
In conclusion:
- For no solution, 'a' must satisfy a^2 - 12 ≠ 0.
- For exactly one solution, 'a' must satisfy a^2 - 12 = 0 and a + 4 ≠ 0.
- For infinitely many solutions, 'a' must satisfy a^2 - 12 = 0 and a + 4 = 0.
To determine the number of solutions for the given system of linear equations, we need to convert the system into an augmented matrix and then row reduce it using Gaussian elimination.
First, let's write the system of equations in matrix form:
```
[1 2 -3 | 4]
[3 -1 5 | 2]
[4 1 (a^2-14) | a+2]
```
Now, we perform row reduction:
Step 1: Swap R1 and R2 to simplify calculations.
```
[3 -1 5 | 2]
[1 2 -3 | 4]
[4 1 (a^2-14) | a+2]
```
Step 2: Perform R2 => (R2 - 3*R1) and R3 => (R3 - 4*R1).
```
[3 -1 5 | 2]
[0 5 -18 | 10]
[0 5 (a^2-34) | a-6]
```
Step 3: Swap R2 and R3 to have a better arrangement for further calculations.
```
[3 -1 5 | 2]
[0 5 (a^2-34) | a-6]
[0 5 -18 | 10]
```
Step 4: Perform R3 => (R3 - R2).
```
[3 -1 5 | 2]
[0 5 (a^2-34) | a-6]
[0 0 ((a^2-34)+18) | (10-(a-6))]
```
Simplifying the last row gives us:
```
[3 -1 5 | 2]
[0 5 (a^2-34) | a-6]
[0 0 (a^2-16) | (16-a)]
```
Now, let's analyze the last row:
Case 1: (a^2-16) = 0
In this case, the system will have no solutions since there would be a row of zeroes that represents an inconsistent equation.
Case 2: (a^2-16) ≠ 0
In this case, the system will have a unique solution because all the variables are represented without any contradictions.
Therefore, the system will have:
- No solutions if (a^2-16) = 0.
- Exactly one solution if (a^2-16) ≠ 0.
- Infinitely many solutions are not possible for this system since there are no free variables.
To summarize:
- For (a^2-16) = 0, there will be no solutions.
- For (a^2-16) ≠ 0, there will be exactly one solution.
d =
| 1 2 -3 |
|3 -1 5 |
| 4 1 (a^2-14)|
d = -(a^2-14)-6(a^2-14)+40 - 9-12-5
d = -7(a^2-14)+14
d = -7a^2 + 8*14 = (-a^2 +16)7
if d is zero, all solutions have 0 in the denominator
so if a - 4 or -4, solution is undefined
Now if a is such that equation 3 is a multiple of one of the first two, there is no unique solution (lines are parallel)
now solutions are of form:
P/d
where P is the determinant of the matrix resulting from replacing row one with the right side for x, row 2 for the right for y, row 3 for the right for z
so for z for example
| +1 +2 +4 |
| +3 -1 +2 |
| +4 +1 (a+2)|
--------------------
(-a^2 +16)7
[-(a+2)+16+12+16-2 -6(a+2) ] /(-a^2 +16)7
[-7(a+2)+42]/(-a^2 +16)7
[-7a + 28 ] /(-a^2 +16)7
-7 [ a-4 ] / -7[a^2-16]
1/(a+4)
hmmm, z is undefined if a = -4
etc