Question 174262

{{{x+4y=7}}} Start with the given equation.



{{{4y=7-x}}} Subtract {{{x}}} from both sides.



{{{4y=-x+7}}} Rearrange the terms.



{{{y=(-x+7)/(4)}}} Divide both sides by {{{4}}} to isolate y.



{{{y=((-1)/(4))x+(7)/(4)}}} Break up the fraction.



{{{y=-(1/4)x+7/4}}} Reduce.



We can see that the equation {{{y=-(1/4)x+7/4}}} has a slope {{{m=-1/4}}} and a y-intercept {{{b=7/4}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=-1/4}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=-1/4}}}  and the coordinates of the given point *[Tex \LARGE \left\(4,8\right\)].



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



{{{y-8=(-1/4)(x-4)}}} Plug in {{{m=-1/4}}}, {{{x[1]=4}}}, and {{{y[1]=8}}}



{{{y-8=(-1/4)x+(-1/4)(-4)}}} Distribute



{{{y-8=(-1/4)x+1}}} Multiply



{{{y=(-1/4)x+1+8}}} Add 8 to both sides. 



{{{y=(-1/4)x+9}}} Combine like terms. 



So the equation of the line parallel to {{{x+4y=7}}} that goes through the point *[Tex \LARGE \left\(4,8\right\)] is {{{y=(-1/4)x+9}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,-(1/4)x+7/4,(-1/4)x+9),
circle(4,8,0.08),
circle(4,8,0.10),
circle(4,8,0.12))}}}Graph of the original equation {{{y=-(1/4)x+7/4}}} (red) and the parallel line {{{y=(-1/4)x+9}}} (green) through the point *[Tex \LARGE \left\(4,8\right\)].