Question 243547

{{{2x-7y=3}}} Start with the given equation.



{{{-7y=3-2x}}} Subtract {{{2x}}} from both sides.



{{{-7y=-2x+3}}} Rearrange the terms.



{{{y=(-2x+3)/(-7)}}} Divide both sides by {{{-7}}} to isolate y.



{{{y=((-2)/(-7))x+(3)/(-7)}}} Break up the fraction.



{{{y=(2/7)x-3/7}}} Reduce.



We can see that the equation {{{y=(2/7)x-3/7}}} has a slope {{{m=2/7}}} and a y-intercept {{{b=-3/7}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=2/7}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=2/7}}}  and the coordinates of the given point *[Tex \LARGE \left\(4,-4\right\)].



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



{{{y--4=(2/7)(x-4)}}} Plug in {{{m=2/7}}}, {{{x[1]=4}}}, and {{{y[1]=-4}}}



{{{y+4=(2/7)(x-4)}}} Rewrite {{{y--4}}} as {{{y+4}}}



{{{y+4=(2/7)x+(2/7)(-4)}}} Distribute



{{{y+4=(2/7)x-8/7}}} Multiply



{{{y=(2/7)x-8/7-4}}} Subtract 4 from both sides. 



{{{y=(2/7)x-36/7}}} Combine like terms. 



So the equation of the line parallel to {{{2x-7y=3}}} that goes through the point *[Tex \LARGE \left\(4,-4\right\)] is {{{y=(2/7)x-36/7}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,(2/7)x-3/7,(2/7)x-36/7),
circle(4,-4,0.08),
circle(4,-4,0.10),
circle(4,-4,0.12))}}}

Graph of the original equation {{{y=(2/7)x-3/7}}} (red) and the parallel line {{{y=(2/7)x-36/7}}} (green) through the point *[Tex \LARGE \left\(4,-4\right\)].