Question 257320


{{{x+6y=7}}} Start with the given equation.



{{{6y=7-x}}} Subtract {{{x}}} from both sides.



{{{6y=-x+7}}} Rearrange the terms.



{{{y=(-x+7)/(6)}}} Divide both sides by {{{6}}} to isolate y.



{{{y=((-1)/(6))x+(7)/(6)}}} Break up the fraction.



{{{y=-(1/6)x+7/6}}} Reduce.



We can see that the equation {{{y=-(1/6)x+7/6}}} has a slope {{{m=-1/6}}} and a y-intercept {{{b=7/6}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=-1/6}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=-1/6}}}  and the coordinates of the given point *[Tex \LARGE \left\(6,9\right\)].



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



{{{y-9=(-1/6)(x-6)}}} Plug in {{{m=-1/6}}}, {{{x[1]=6}}}, and {{{y[1]=9}}}



{{{y-9=(-1/6)x+(-1/6)(-6)}}} Distribute



{{{y-9=(-1/6)x+1}}} Multiply



{{{y=(-1/6)x+1+9}}} Add 9 to both sides. 



{{{y=(-1/6)x+10}}} Combine like terms. 



So the equation of the line parallel to {{{x+6y=7}}} that goes through the point *[Tex \LARGE \left\(6,9\right\)] is {{{y=(-1/6)x+10}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(1/6)x+7/6,(-1/6)x+10),
circle(6,9,0.08),
circle(6,9,0.10),
circle(6,9,0.12))}}}

Graph of the original equation {{{y=-(1/6)x+7/6}}} (red) and the parallel line {{{y=(-1/6)x+10}}} (green) through the point *[Tex \LARGE \left\(6,9\right\)].