Question 168152




Since the equation {{{y=3x+4}}} is in slope intercept form {{{y=mx+b}}}, this means that the equation has the slope {{{m=3}}} and the y-intercept {{{b=4}}}.



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\(3,2\right\)].



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



{{{y-2=3(x-3)}}} Plug in {{{m=3}}}, {{{x[1]=3}}}, and {{{y[1]=2}}}



{{{y-2=3x+3(-3)}}} Distribute



{{{y-2=3x-9}}} Multiply



{{{y=3x-9+2}}} Add 2 to both sides. 



{{{y=3x-7}}} Combine like terms. 



So the equation of the line parallel to {{{y=3x+4}}} that goes through the point *[Tex \LARGE \left\(3,2\right\)] is {{{y=3x-7}}}.



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+4,3x-7),
circle(3,2,0.08),
circle(3,2,0.10),
circle(3,2,0.12))}}}Graph of the original equation {{{y=3x+4}}} (red) and the parallel line {{{y=3x-7}}} (green) through the point *[Tex \LARGE \left\(3,2\right\)].