Question 243670
# 1




We can see that the equation {{{y=4x-6}}} has a slope {{{m=4}}} and a y-intercept {{{b=-6}}}.



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

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=4}}}  and the coordinates of the given point *[Tex \LARGE \left\(0,6\right\)].



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



{{{y-6=4(x-0)}}} Plug in {{{m=4}}}, {{{x[1]=0}}}, and {{{y[1]=6}}}



{{{y-6=4x+4(-0)}}} Distribute



{{{y-6=4x+0}}} Multiply



{{{y=4x+0+6}}} Add 6 to both sides. 



{{{y=4x+6}}} Combine like terms. 



So the equation of the line parallel to {{{y=4x-6}}} that goes through the point *[Tex \LARGE \left\(0,6\right\)] is {{{y=4x+6}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,4x-6,4x+6),
circle(0,6,0.08),
circle(0,6,0.10),
circle(0,6,0.12))}}}

Graph of the original equation {{{y=4x-6}}} (red) and the parallel line {{{y=4x+6}}} (green) through the point *[Tex \LARGE \left\(0,6\right\)]. 



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# 2





If you want to find the equation of line with a given a slope of {{{-8}}} which goes through the point (6,9), you can simply use the point-slope formula to find the equation:



---Point-Slope Formula---
{{{y-y[1]=m(x-x[1])}}} where {{{m}}} is the slope, and *[Tex \Large \left(x_{1},y_{1}\right)] is the given point



So lets use the Point-Slope Formula to find the equation of the line



{{{y-9=-8(x-6)}}} Plug in {{{m=-8}}}, {{{x[1]=6}}}, and {{{y[1]=9}}} (these values are given)



Now if you just want the equation in point slope form, then the answer is simply {{{y-9=-8(x-6)}}}. However, if you want it in slope-intercept form, then read on...



{{{y-9=-8x+(-8)(-6)}}} Distribute {{{-8}}}



{{{y-9=-8x+48}}} Multiply {{{-8}}} and {{{-6}}} to get {{{48}}}



{{{y=-8x+48+9}}} Add 9 to  both sides to isolate y



{{{y=-8x+57}}} Add


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Answer:



So the equation of the line with a slope of {{{-8}}} which goes through the point (6,9) is:


{{{y=-8x+57}}} which is now in {{{y=mx+b}}} form where the slope is {{{m=-8}}} and the y-intercept is {{{b=57}}}