Question 924054
given:
the pair of linear equations 
{{{3x + 7y = k}}} 
{{{12x + 2ky = 4k + 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 + 7y = k}}}

{{{7y =k-3x }}}

{{{y =-(3/7)x +k/7}}} => slope is {{{m[1]=-(3/7)}}}



{{{12x + 2ky = 4k + 1 }}}

{{{2ky = -12x +4k + 1 }}}

{{{y = -(12/2k)x +4k/2k + 1/2k }}}

{{{y = -(6/k)x +2 + 1/2k }}}

=> slope is {{{m[2]=  -(6/k)}}}

now, since {{{m[1]=m[2]}}}, we have

{{{-(3/7)= -(6/k)}}} -solve for {{{k}}}
{{{-3k= -6*7}}} ...both sides multiply by {{{-1}}} 

{{{3k=42}}}

{{{k= 42 /3}}}

{{{k=14}}}

then your system is:


{{{3x + 7y =14}}}
{{{12x + 2*14y = 4*14 + 1 }}}


{{{3x + 7y =14}}}
{{{12x + 28y = 57}}}
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see the graph: 

{{{ graph( 600, 600, -30,30, -30, 30, -(3/7)x+14,  -(6/14)x +2 + 1/28) }}}