Question 128458
# 1



{{{12x+3y=6}}} Start with the given equation



{{{3y=6-12x}}}  Subtract {{{12 x}}} from both sides



{{{3y=-12x+6}}} Rearrange the equation



{{{y=(-12x+6)/(3)}}} Divide both sides by {{{3}}}



{{{y=(-12/3)x+(6)/(3)}}} Break up the fraction



{{{y=-4x+2}}} Reduce




So the equation is now in slope-intercept form ({{{y=mx+b}}}) where the slope is {{{m=-4}}} and the y-intercept is {{{b=2}}}



So to get the equation into function form, simply replace y with f(x)




So the equation changes to the function {{{f(x)=-4x+2}}}







Looking at {{{y=-4x+2}}} we can see that the equation is in slope-intercept form {{{y=mx+b}}} where the slope is {{{m=-4}}} and the y-intercept is {{{b=2}}} 



Since {{{b=2}}} this tells us that the y-intercept is *[Tex \LARGE \left(0,2\right)].Remember the y-intercept is the point where the graph intersects with the y-axis


So we have one point *[Tex \LARGE \left(0,2\right)]


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,2,.1)),
  blue(circle(0,2,.12)),
  blue(circle(0,2,.15))
)}}}



Now since the slope is comprised of the "rise" over the "run" this means

{{{slope=rise/run}}}


Also, because the slope is {{{-4}}}, this means:


{{{rise/run=-4/1}}}



which shows us that the rise is -4 and the run is 1. This means that to go from point to point, we can go down 4  and over 1




So starting at *[Tex \LARGE \left(0,2\right)], go down 4 units 

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,2,.1)),
  blue(circle(0,2,.12)),
  blue(circle(0,2,.15)),
  blue(arc(0,2+(-4/2),2,-4,90,270))
)}}}


and to the right 1 unit to get to the next point *[Tex \LARGE \left(1,-2\right)]

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,2,.1)),
  blue(circle(0,2,.12)),
  blue(circle(0,2,.15)),
  blue(circle(1,-2,.15,1.5)),
  blue(circle(1,-2,.1,1.5)),
  blue(arc(0,2+(-4/2),2,-4,90,270)),
  blue(arc((1/2),-2,1,2, 0,180))
)}}}



Now draw a line through these points to graph {{{y=-4x+2}}}


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  graph(500,500,-10,10,-10,10,-4x+2),
  blue(circle(0,2,.1)),
  blue(circle(0,2,.12)),
  blue(circle(0,2,.15)),
  blue(circle(1,-2,.15,1.5)),
  blue(circle(1,-2,.1,1.5)),
  blue(arc(0,2+(-4/2),2,-4,90,270)),
  blue(arc((1/2),-2,1,2, 0,180))
)}}} So this is the graph of {{{y=-4x+2}}} through the points *[Tex \LARGE \left(0,2\right)] and *[Tex \LARGE \left(1,-2\right)]




<hr>



# 2






{{{4x-2y=16}}} Start with the given equation



{{{-2y=16-4x}}}  Subtract {{{4 x}}} from both sides



{{{-2y=-4x+16}}} Rearrange the equation



{{{y=(-4x+16)/(-2)}}} Divide both sides by {{{-2}}}



{{{y=(-4/-2)x+(16)/(-2)}}} Break up the fraction



{{{y=2x-8}}} Reduce




So the equation is now in slope-intercept form ({{{y=mx+b}}}) where the slope is {{{m=2}}} and the y-intercept is {{{b=-8}}}




So to get the equation into function form, simply replace y with f(x)




So the equation changes to the function {{{f(x)=2x-8}}}






Looking at {{{y=2x-8}}} we can see that the equation is in slope-intercept form {{{y=mx+b}}} where the slope is {{{m=2}}} and the y-intercept is {{{b=-8}}} 



Since {{{b=-8}}} this tells us that the y-intercept is *[Tex \LARGE \left(0,-8\right)].Remember the y-intercept is the point where the graph intersects with the y-axis


So we have one point *[Tex \LARGE \left(0,-8\right)]


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,-8,.1)),
  blue(circle(0,-8,.12)),
  blue(circle(0,-8,.15))
)}}}



Now since the slope is comprised of the "rise" over the "run" this means

{{{slope=rise/run}}}


Also, because the slope is {{{2}}}, this means:


{{{rise/run=2/1}}}



which shows us that the rise is 2 and the run is 1. This means that to go from point to point, we can go up 2  and over 1




So starting at *[Tex \LARGE \left(0,-8\right)], go up 2 units 

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,-8,.1)),
  blue(circle(0,-8,.12)),
  blue(circle(0,-8,.15)),
  blue(arc(0,-8+(2/2),2,2,90,270))
)}}}


and to the right 1 unit to get to the next point *[Tex \LARGE \left(1,-6\right)]

{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  blue(circle(0,-8,.1)),
  blue(circle(0,-8,.12)),
  blue(circle(0,-8,.15)),
  blue(circle(1,-6,.15,1.5)),
  blue(circle(1,-6,.1,1.5)),
  blue(arc(0,-8+(2/2),2,2,90,270)),
  blue(arc((1/2),-6,1,2, 180,360))
)}}}



Now draw a line through these points to graph {{{y=2x-8}}}


{{{drawing(500,500,-10,10,-10,10,
  grid(1),
  graph(500,500,-10,10,-10,10,2x-8),
  blue(circle(0,-8,.1)),
  blue(circle(0,-8,.12)),
  blue(circle(0,-8,.15)),
  blue(circle(1,-6,.15,1.5)),
  blue(circle(1,-6,.1,1.5)),
  blue(arc(0,-8+(2/2),2,2,90,270)),
  blue(arc((1/2),-6,1,2, 180,360))
)}}} So this is the graph of {{{y=2x-8}}} through the points *[Tex \LARGE \left(0,-8\right)] and *[Tex \LARGE \left(1,-6\right)]