Question 1085335
For the system in the problem to be inconsistent,
the coefficients of x and y in {{{(2k-1)x +(k-1)y=2k+1}}}
have to equal to the coefficients in {{{4x+y=3}}}  multiplied by a certain number,
but the same cannot be true of the terms not connected to variables:
{{{2k+1}}} and {{{3}}} .
So, it must be that {{{(2k-1)/4=(k-1)/1}}} .
The ratio is {{{k-1}}} ,
but that ratio must be different from {{{(2k+1)/3}}}
Solving:
{{{(2k-1)/4=(k-1)/1}}}
{{{2k-1=4(k-1)}}}
{{{2k-1=4k-4}}}
{{{-1+4=4k-2k}}}
{{{3=2k}}}
{{{3/2=k}}}
With {{{k=3/2}}} , the coefficients' ratio is {{{k-1=3/2-1=1/2}}} ,
and {{{(2k+1)/3=(2("3 / 2")+1)/3=(3+1)/3=4/3}}} is not the same.
So, {{{highlight(k=3/2)}}} makes the system inconsistent.
 
LONG-WINDED EXPLANATION:
Linear equations graph as straight lines.
If the lines intersect, there is a solution to the system:
the set of coordinates for the intersection point.
Lines that intersect do not have the same slope.
The lines could have the same slope,
as in {{{system(4x+y=3,4x+y=5)}}} , or {{{system(4x+y=3,4x+y=3)}}} , or {{{system(4x+y=3,8x+2y=6)}}} , or {{{system(4x+y=3,8x+2y=10)}}} .
If the slopes are the same it could be that the lines are really the same line,
as is the case with {{{4x+y=3}}} and {{{8x+2y=6}}} ,
and in that case, there is an infinite number of solutions:
the sets of coordinates for each of the infinite number of points on the line.
The other possibility, if the slopes are the same is cases like {{{system(4x+y=3,4x+y=5)}}} ,
where it is obvious that the two equations cannot be true at the same time.
There are no solutions, and the system of linear equations is called inconsistent.
If {{{4x+y}}} is {{{3}}} it cannot also be {{{5}}} for the same x and y values;
one equation is not consistent with the other.
Those equations represent parallel lines, with no point in common.
It is the same case with {{{system(4x+y=3,8x+2y=10)}}} ,
although it is not so obvious. {{{8x+2y=10}}} is equivalent to {{{4x+y=5}}} .
The coefficients of {{{8x+2y=10}}} are 2 times those of {{{4x+y=3}}} ,
but the number 10 (not attached to any variable, on the other side of the equal sign)
is not 2 times 3.