SOLUTION: 4x+y=3 and (2k-1)x +(k-1)y=2k+1 is inconsistent ,find k

Question 1085335: 4x+y=3 and (2k-1)x +(k-1)y=2k+1 is inconsistent ,find k
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
For the system in the problem to be inconsistent,
the coefficients of x and y in
have to equal to the coefficients in multiplied by a certain number,
but the same cannot be true of the terms not connected to variables:
and .
So, it must be that .
The ratio is ,
but that ratio must be different from
Solving:

With , the coefficients' ratio is ,
and is not the same.
So, 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 , or , or , or .
If the slopes are the same it could be that the lines are really the same line,
as is the case with and ,
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 ,
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 is it cannot also be 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 ,
although it is not so obvious. is equivalent to .
The coefficients of are 2 times those of ,
but the number 10 (not attached to any variable, on the other side of the equal sign)
is not 2 times 3.
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