SOLUTION: the pair of linear equations 3x + 7y = k, 12x + 2ky = 4k + 1 do not have any solutions if k = ?

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Question 924054: the pair of linear equations 3x + 7y = k, 12x + 2ky = 4k + 1 do not have any solutions if k = ?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
given:
the pair of linear equations
3x+%2B+7y+=+k
12x+%2B+2ky+=+4k+%2B+1+
find: k such that the pair of linear equations does not have any solutions
recall:
if solving a linear system, in fact there are only three possible cases:
No solution
One solution
Infinitely many solutions
When there is no+solution the equations are called "inconsistent" and the lines are parallel
One or infinitely many solutions are called "consistent"-overlapping lines

so, your lines must be parallel, and they will be parallel if they have same+slope
now, solve your line for y+
3x+%2B+7y+=+k
7y+=k-3x+
y+=-%283%2F7%29x+%2Bk%2F7 => slope is m%5B1%5D=-%283%2F7%29


12x+%2B+2ky+=+4k+%2B+1+
2ky+=+-12x+%2B4k+%2B+1+
y+=+-%2812%2F2k%29x+%2B4k%2F2k+%2B+1%2F2k+
y+=+-%286%2Fk%29x+%2B2+%2B+1%2F2k+
=> slope is m%5B2%5D=++-%286%2Fk%29
now, since m%5B1%5D=m%5B2%5D, we have
-%283%2F7%29=+-%286%2Fk%29 -solve for k
-3k=+-6%2A7 ...both sides multiply by -1
3k=42
k=+42+%2F3
k=14
then your system is:

3x+%2B+7y+=14
12x+%2B+2%2A14y+=+4%2A14+%2B+1+

3x+%2B+7y+=14
12x+%2B+28y+=+57
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see the graph: