Question 253688


We can see that the equation {{{y=-(3/4)x+1}}} has a slope {{{m=-3/4}}} and a y-intercept {{{b=1}}}.



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

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=-3/4}}}  and the coordinates of the given point *[Tex \LARGE \left\(4,-6\right\)].



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



{{{y--6=-(3/4)(x-4)}}} Plug in {{{m=-3/4}}}, {{{x[1]=4}}}, and {{{y[1]=-6}}}



{{{y+6=-(3/4)(x-4)}}} Rewrite {{{y--6}}} as {{{y+6}}}



{{{y+6=-(3/4)x-(3/4)(-4)}}} Distribute



{{{y+6=-(3/4)x+3}}} Multiply



{{{y=-(3/4)x+3-6}}} Subtract 6 from both sides. 



{{{y=-(3/4)x-3}}} Combine like terms. 



So the equation of the line parallel to {{{y=-3/4x+1}}} that goes through the point *[Tex \LARGE \left\(4,-6\right\)] is {{{y=-(3/4)x-3}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(3/4)x+1,-(3/4)x-3),
circle(4,-6,0.08),
circle(4,-6,0.10),
circle(4,-6,0.12))}}}


Graph of the original equation {{{y=-(3/4)x+1}}} (red) and the parallel line {{{y=-(3/4)x-3}}} (green) through the point *[Tex \LARGE \left\(4,-6\right\)].