Question 333866
Note: {{{y=3x-7}}} is the same as {{{f(x)=3x-7}}}



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\(0,4\right\)].



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



{{{y-4=3(x-0)}}} Plug in {{{m=3}}}, {{{x[1]=0}}}, and {{{y[1]=4}}}



{{{y-4=3x+3(-0)}}} Distribute



{{{y-4=3x+0}}} Multiply



{{{y=3x+0+4}}} Add 4 to both sides. 



{{{y=3x+4}}} Combine like terms. 



So the equation of the line parallel to {{{y=3x-7}}} that goes through the point *[Tex \LARGE \left\(0,4\right\)] is {{{y=3x+4}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,3x-7,3x+4),
circle(0,4,0.08),
circle(0,4,0.10),
circle(0,4,0.12))}}} Graph of the original equation {{{y=3x-7}}} (red) and the parallel line {{{y=3x+4}}} (green) through the point *[Tex \LARGE \left\(0,4\right\)].