Question 376310


{{{8x-9y=5}}} Start with the given equation.



{{{-9y=5-8x}}} Subtract {{{8x}}} from both sides.



{{{-9y=-8x+5}}} Rearrange the terms.



{{{y=(-8x+5)/(-9)}}} Divide both sides by {{{-9}}} to isolate y.



{{{y=((-8)/(-9))x+(5)/(-9)}}} Break up the fraction.



{{{y=(8/9)x-5/9}}} Reduce.



We can see that the equation {{{y=(8/9)x-5/9}}} has a slope {{{m=8/9}}} and a y-intercept {{{b=-5/9}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=8/9}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=8/9}}}  and the coordinates of the given point *[Tex \LARGE \left\(9,-9\right\)].



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



{{{y--9=(8/9)(x-9)}}} Plug in {{{m=8/9}}}, {{{x[1]=9}}}, and {{{y[1]=-9}}}



{{{y+9=(8/9)(x-9)}}} Rewrite {{{y--9}}} as {{{y+9}}}



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



{{{y+9=(8/9)x-8}}} Multiply



{{{y=(8/9)x-8-9}}} Subtract 9 from both sides. 



{{{y=(8/9)x-17}}} Combine like terms. 



So the equation of the line parallel to {{{8x-9y=5}}} that goes through the point *[Tex \LARGE \left\(9,-9\right\)] is {{{y=(8/9)x-17}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,(8/9)x-5/9,(8/9)x-17),
circle(9,-9,0.08),
circle(9,-9,0.10),
circle(9,-9,0.12))}}}Graph of the original equation {{{y=(8/9)x-5/9}}} (red) and the parallel line {{{y=(8/9)x-17}}} (green) through the point *[Tex \LARGE \left\(9,-9\right\)]. 



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Jim