Question 336496

We can see that the equation {{{y=4x+5}}} has a slope {{{m=4}}} and a y-intercept {{{b=5}}}.



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

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=4}}}  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=4(x--3)}}} Plug in {{{m=4}}}, {{{x[1]=-3}}}, and {{{y[1]=-2}}}



{{{y--2=4(x+3)}}} Rewrite {{{x--3}}} as {{{x+3}}}



{{{y+2=4(x+3)}}} Rewrite {{{y--2}}} as {{{y+2}}}



{{{y+2=4x+4(3)}}} Distribute



{{{y+2=4x+12}}} Multiply



{{{y=4x+12-2}}} Subtract 2 from both sides. 



{{{y=4x+10}}} Combine like terms. 



So the equation of the line parallel to {{{y=4x+5}}} that goes through the point *[Tex \LARGE \left\(-3,-2\right\)] is {{{y=4x+10}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,4x+5,4x+10),
circle(-3,-2,0.08),
circle(-3,-2,0.10),
circle(-3,-2,0.12))}}} Graph of the original equation {{{y=4x+5}}} (red) and the parallel line {{{y=4x+10}}} (green) through the point *[Tex \LARGE \left\(-3,-2\right\)]. 



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Jim