Question 153218


{{{3x-y=7}}} Start with the given equation.



{{{-y=7-3x}}} Subtract {{{3x}}} from both sides.



{{{-y=-3x+7}}} Rearrange the terms.



{{{y=(-3x+7)/(-1)}}} Divide both sides by {{{-1}}} to isolate y.



{{{y=((-3)/(-1))x+(7)/(-1)}}} Break up the fraction.



{{{y=3x-7}}} Reduce.



We can see that the equation {{{y=3x-7}}} has a slope {{{m=3}}} and a y-intercept {{{b=-7}}}.



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

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



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



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



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



{{{y-3=3x-12}}} Multiply



{{{y=3x-12+3}}} Add 3 to both sides. 



{{{y=3x-9}}} Combine like terms. 



So the equation of the line parallel to {{{3x-y=7}}} that goes through the point *[Tex \LARGE \left\(4,3\right\)] is {{{y=3x-9}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,3*x-7,3x-9),
circle(4,3,0.08),
circle(4,3,0.10),
circle(4,3,0.12))}}}Graph of the original equation {{{y=3*x-7}}} (red) and the parallel line {{{y=3x-9}}} (green) through the point *[Tex \LARGE \left\(4,3\right\)].