Question 175881
Note: the origin is the point (0,0)



{{{-4x + 3y= 12}}} Start with the given equation.



{{{3y=12+4x}}} Add {{{4x}}} to both sides.



{{{3y=4x+12}}} Rearrange the terms.



{{{y=(4x+12)/(3)}}} Divide both sides by {{{3}}} to isolate y.



{{{y=((4)/(3))x+(12)/(3)}}} Break up the fraction.



{{{y=(4/3)x+4}}} Reduce.



We can see that the equation {{{y=(4/3)x+4}}} has a slope {{{m=4/3}}} and a y-intercept {{{b=4}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=4/3}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=4/3}}}  and the coordinates of the given point *[Tex \LARGE \left\(0,0\right\)].



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-0=(4/3)(x-0)}}} Plug in {{{m=4/3}}}, {{{x[1]=0}}}, and {{{y[1]=0}}}



{{{y=(4/3)(x-0)}}} Simplify



{{{y=(4/3)x+(4/3)(0)}}} Distribute



{{{y=(4/3)x+0}}} Multiply



{{{y=(4/3)x}}} Simplify




So the equation of the line parallel to {{{-4x + 3y= 12}}} that goes through the point *[Tex \LARGE \left\(0,0\right\)] is {{{y=(4/3)x}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,(4/3)x+4,(4/3)x),
circle(0,0,0.08),
circle(0,0,0.10),
circle(0,0,0.12))}}}Graph of the original equation {{{y=(4/3)x+4}}} (red) and the parallel line {{{y=(4/3)x}}} (green) through the point *[Tex \LARGE \left\(0,0\right\)].