Question 1206698
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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 {{{k <> 0}}} and {{{k <> -4}}}
These restrictions on k are to avoid division by zero errors in the matrix row reduction shown below.
<table border = "1" cellpadding = "5"><tr><td>k</td><td>1</td><td>-2</td></tr><tr><td>4</td><td>-1</td><td>2</td></tr></table>


<table border = "1" cellpadding = "5"><tr><td>1</td><td>1/k</td><td>-2/k</td><td>(1/k)*R1 --> R1</td></tr><tr><td>4</td><td>-1</td><td>2</td><td></td></tr></table>


<table border = "1" cellpadding = "5"><tr><td>1</td><td>1/k</td><td>-2/k</td><td></td></tr><tr><td>0</td><td>-(k+4)/k</td><td>(2k+8)/k</td><td>R2 - 4R1 --> R2</td></tr></table>


<table border = "1" cellpadding = "5"><tr><td>1</td><td>1/k</td><td>-2/k</td><td></td></tr><tr><td>0</td><td>1</td><td>-2</td><td>(-k/(k+4))*R2 --> R2</td></tr></table>


<table border = "1" cellpadding = "5"><tr><td>1</td><td>0</td><td>0</td><td>R1 - (1/k)*R2 --> R1</td></tr><tr><td>0</td><td>1</td><td>-2</td><td></td></tr></table>
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
{{{system(kx+y = -2,4x-y=2)}}}
to this system
{{{system(-4x+y = -2,4x-y=2)}}}
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 {{{system(kx+y=-2,4x-y=2)}}} 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. 
<a href="https://www.desmos.com/calculator/mh8pmourgs">https://www.desmos.com/calculator/mh8pmourgs</a>
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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