SOLUTION: Consider the system of equations x + 2y - 3z = a 2x + 6y - 11z = b x - 2y + 7z = c where a, b and c are three real numbers. 1 What relation must the parameters a, b and c sati

Algebra ->  College  -> Linear Algebra -> SOLUTION: Consider the system of equations x + 2y - 3z = a 2x + 6y - 11z = b x - 2y + 7z = c where a, b and c are three real numbers. 1 What relation must the parameters a, b and c sati      Log On


   

Question 1204100: Consider the system of equations
x + 2y - 3z = a
2x + 6y - 11z = b
x - 2y + 7z = c
where a, b and c are three real numbers.
1 What relation must the parameters a, b and c satisfy for the system of equations to have at least
the system of equations has at least one solution?
2. Assuming that a, b and c satisfy the relation that allows the system to have at least one solution, what relation must a, b and c satisfy?
the system has at least one solution, calculate in function of a, b and c, the general solution of the system of equations using Gauss's method.
Can the above linear system have a unique solution in R3?
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Consider the system of equations
+x+%2B+2y+-+3z+=+a
+2x+%2B+6y+-+11z+=+b
+x+-+2y+%2B+7z+=+c
where +a , +b+ and +c+ are three real numbers.


Solution:
+x+%2B+2y+-+3z+=+a
+2x+%2B+6y+-+11z+=+b
+x+-+2y+%2B+7z+=+c

Δ=+matrix%283%2C3%2C%0D%0A1%2C2%2C-3%2C%0D%0A2%2C6%2C-11%2C%0D%0A1%2C-2%2C7%29 =0

Δ1=+matrix%283%2C3%2C%0D%0Aa%2C+2%2C+-3%2C%0D%0Ab%2C+6%2C-11%2C%0D%0Ac%2C+-2%2C7%29 =4%285a-2b-c%29

Δ2=+matrix%283%2C3%2C%0D%0A1%2C+a%2C-3%2C%0D%0A2%2Cb%2C-11%2C%0D%0A1%2Cc%2C7%29 =-5%285a-2b-c%29+

Δ3=+matrix%283%2C3%2C%0D%0A1%2C2%2Ca%2C%0D%0A2%2C6%2Cb%2C%0D%0A1%2C-2%2Cc%29 =-2%285a-2b-c%29+

If +5a=2b%2Bc+ => Δ1=Δ2=Δ3=0 => system will have infinitely many solutions .


Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll convert the system of equations into a matrix as shown below.
Then I'll get that matrix into row echelon form (REF)

Each matrix is presented as a table. In my opinion, the grid lines help separate things to make the entries look more cleaner.
Normally however, matrix notation will not have these helpful grid lines.
12-3a
26-11b
1-27c

12-3a
26-11b
0-410c-aR3 - R1 --> R3

Notation like R3 - R1 --> R3 means we subtract rows 3 and 1, and store the results into row 3.
12-3a
02-5b-2aR2 - 2*R1 --> R2
0-410c-a

12-3a
01-5/2(b-2a)/2(1/2)*R2 --> R2
0-410c-a

12-3a
01-5/2(b-2a)/2
000-5a+2b+cR3 + 4*R2 --> R3

We have gone from the matrix %28matrix%283%2C4%2C1%2C2%2C-3%2Ca%2C2%2C6%2C-11%2Cb%2C1%2C-2%2C7%2Cc%29%29 to the matrix

The last row leads to the equation
0x+0y+0z = -5a+2b+c
or
0 = -5a+2b+c

Solve for c to get
c = 5a-2b

This will mean we have infinitely many solutions if and only if c = 5a-2b
Otherwise, -5a+2b+c is nonzero and it causes a contradiction, and hence no solutions.

The values of 'a' and b can be any two real numbers you want.

---------------------------------------------------

Let's look at an example that has infinitely many solutions.
a = 1, b = 2
c = 5a-2b = 5*1-2*2 = 1
Therefore this system
system%28x%2B2y-3z=1%2C2x%2B6y-11z=2%2Cx-2y%2B7z=1%29
is consistent and dependent.
It has infinitely many solutions.
I'll let the student verify this claim, and the later claims mentioned below.

Another example with infinitely many solutions.
a = 5, b = 3
c = 5a-2b = 5*5-2*3 = 19
This system
system%28x%2B2y-3z=5%2C2x%2B6y-11z=3%2Cx-2y%2B7z=19%29
is consistent and dependent.
It has infinitely many solutions.

Let's look at an example of a system that has no solutions.
a = 1, b = 2, c = 3
These a,b,c values do not satisfy the equation c = 5a-2b
Therefore this system shown below has no solutions (it is inconsistent)
system%28x%2B2y-3z=1%2C2x%2B6y-11z=2%2Cx-2y%2B7z=3%29


As you can see, we cannot pick a trio of a,b,c values to have the system produce exactly one unique solution.
Either we have infinitely many solutions, or none at all.