Question 1126889


both lines should have the same slope and y−intercept
 they are one line, coincident, meaning they have all points in common
 this means that there are an infinite number of solutions to the system


{{{-8x -6y = h}}}
{{{12x + ky = 24}}}
-------------------

first write both equations in slope-intercept form:

{{{-8x -6y = h}}}

{{{-8x -h = 6y}}}

{{{y=-(8/6)x -h/6 }}}

{{{y=-(4/3)x -h/6}}} ............eq.1=> slope is {{{-(4/3)}}} and y-intercept is {{{-h/6}}}


{{{12x + ky = 24}}}

{{{ky = 24-12x }}}

{{{y = -(12/k)x +24/k}}}.............eq.2 => slope is {{{-(12/k)}}} and y-intercept is{{{ 24/k}}}


equal slopes:

{{{-(4/3)=-(12/k)}}} ...solve for {{{k}}}; both sides multiply by {{{-1}}}

{{{4/3=12/k}}}........cross multiply

{{{4k=12*3}}}

{{{4k=36}}}

{{{k=9}}}

equal y-intercepts:

{{{-h/6=24/k }}}........substitute {{{k}}}

{{{-h/6=24/9}}} ......cross multiply

{{{-9h=24*6}}}

{{{h=144/-9}}}

{{{h=-16}}}


so, your lines are:

{{{y=-(4/3)x -(-16)/6}}} ->{{{y=-(4/3)x +8/3}}}

{{{y = -(12/9)x +24/9}}}->{{{y = -(4/3)x +8/3}}}


{{{ graph(600, 600, -10,10, -10, 10, -(4/3)x +8/3, -(4/3)x +8/3) }}}