Question 1057709: -kx + y = 3
4x - y = 2
What value of k will make the system consistent? Thank you
Found 3 solutions by MathTherapy, ankor@dixie-net.com, ikleyn:
Answer by MathTherapy(10549)   (Show Source):
You can put this solution on YOUR website! -kx + y = 3
4x - y = 2
What value of k will make the system consistent? Thank you
- kx + y = 3____y = kx + 3 ------ eq (i)
4x - y = 2______y = 4x - 2 ------ eq (ii)
A consistent system means that the 2 equations' slopes ARE NOT EQUAL.
As seen, the system's slopes are k and 4. As long as  , the system will be a CONSISTENT one.
Therefore, for the system to be CONSISTENT, the values for k MUST BE all REALS, except 4.
Answer by ankor@dixie-net.com(22740)  (Show Source):
Answer by ikleyn(52754)  (Show Source):
You can put this solution on YOUR website! .
-kx + y = 3
4x - y = 2
What value of k will make the system consistent? Thank you
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Let's start from the definition.
The system of equations is called consistent if it has a solution.
It doesn't matter whether the solution is unique or there are many solutions.
The system of equations is called consistent if it has at least one solution.
If the system has one, or many, or even INFINITELY many solutions, it is called consistent.
In opposite, if the system doesn't have solutions, it is called inconsistent.
In the Geometry language, the 2x2-system of linear equations is consistent if the two straight lines, representing the equations, EITHER intersect OR coincide.
The 2x2-system of linear equations is inconsistent if the two straight lines, representing the equations, are parallel.
Regarding the given system, the two lines are parallel if and only if
-k = -4, which is equivalent to k = 4.
So, the system is inconsistent if and only if k = 4.
In all other cases the system is CONSISTED.
Answer. The given system is consisted at every value of k different from 4.
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