SOLUTION: Find all values of k for which the given the augmented matrix corresponds to a consistent linear system K 1 -2 4 -1 2

Question 1206698: Find all values of k for which the given the augmented matrix corresponds to a consistent linear system
K 1 -2
4 -1 2
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20849) (Show Source): You can put this solution on YOUR website!

system is:
....eq.1
.....eq.2
----------
...eq.1, solve for
....eq.1a

.....eq.2, solve for
.....eq.2a
from eq.1a and eq.2a we have
....., solve for

....both sides divide by

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

Definition: A consistent system has at least one solution.
In contrast, an inconsistent system has no solutions.

Let's consider a real number k such that and
These restrictions on k are to avoid division by zero errors in the matrix row reduction shown below.

k 1 -2
4 -1 2

1 1/k -2/k (1/k)*R1 --> R1
4 -1 2

1 1/k -2/k
0 -(k+4)/k (2k+8)/k R2 - 4R1 --> R2

1 1/k -2/k
0 1 -2 (-k/(k+4))*R2 --> R2

1 0 0 R1 - (1/k)*R2 --> R1
0 1 -2

The matrix is now in reduced row echelon form (RREF)
The solution is (x,y) = (0,-2) to prove this system is consistent.

Now consider k = 0.
kx+y = -2
0*x+y = -2
y = -2
Then,
4x-y = 2
4x-(-2) = 2
4x+2 = 2
4x = 2-2
4x = 0
x = 0/4
x = 0
We arrive at (x,y) = (0,-2) again.
The system is consistent when k = 0.

Now consider k = -4.
kx+y = -2
-4x+y = -2
We go from this system

to this system

Adding straight down yields 0x+0y = 0 or in short 0 = 0.
This system is consistent when k = -4.
Unlike the other cases, we get infinitely many solutions here. Each solution is of the form (x,y) = (x, 4x-2)
Note x = 0 leads to y = -2 to show that (0,-2) is one of the infinitely many solutions here.

Summary:

We conclude that the system is consistent for any real number k.
Meaning that this system will have at least one solution.
If k = -4 then it has infinitely many solutions of the form (x,4x-2). Otherwise it will have exactly one solution which is (0,-2).

Here is an interactive Desmos graph.
https://www.desmos.com/calculator/mh8pmourgs
Move the slider around for the k value to see the red line rotating around. The center of rotation is (0,-2). When k = -4 the two lines overlap.
It is impossible to pick a value of k to make the system inconsistent.

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