Question 180581



We can see that the equation {{{y=2*x+3}}} has a slope {{{m=2}}} and a y-intercept {{{b=3}}}.



Since parallel lines have equal slopes, this means that we know that the slope of the unknown parallel line is {{{m=2}}}.

Now let's use the point slope formula to find the equation of the parallel line by plugging in the slope {{{m=2}}}  and the coordinates of the given point *[Tex \LARGE \left\(1,1\right\)].



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



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



{{{y-1=2x+2(-1)}}} Distribute



{{{y-1=2x-2}}} Multiply



{{{y=2x-2+1}}} Add 1 to both sides. 



{{{y=2x-1}}} Combine like terms. 



So the equation of the line parallel to {{{-2x+y=3}}} that goes through the point *[Tex \LARGE \left\(1,1\right\)] is {{{y=2x-1}}}.



Here's a graph to visually verify our answer:

{{{drawing(500, 500, -10, 10, -10, 10,
graph(500, 500, -10, 10, -10, 10,2*x+3,2x-1),
circle(1,1,0.08),
circle(1,1,0.10),
circle(1,1,0.12))}}}Graph of the original equation {{{y=2*x+3}}} (red) and the parallel line {{{y=2x-1}}} (green) through the point *[Tex \LARGE \left\(1,1\right\)].