Question 674116


First let's find the slope of the line through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(0,4\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(3,5\right)]. So this means that {{{x[1]=3}}} and {{{y[1]=5}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(0,4\right)].  So this means that {{{x[2]=0}}} and {{{y[2]=4}}}.



{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(4-5)/(0-3)}}} Plug in {{{y[2]=4}}}, {{{y[1]=5}}}, {{{x[2]=0}}}, and {{{x[1]=3}}}



{{{m=(-1)/(0-3)}}} Subtract {{{5}}} from {{{4}}} to get {{{-1}}}



{{{m=(-1)/(-3)}}} Subtract {{{3}}} from {{{0}}} to get {{{-3}}}



{{{m=1/3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(0,4\right)] is {{{m=1/3}}}



Now let's use the point slope formula:



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



{{{y-5=(1/3)(x-3)}}} Plug in {{{m=1/3}}}, {{{x[1]=3}}}, and {{{y[1]=5}}}



{{{y-5=(1/3)x+(1/3)(-3)}}} Distribute



{{{y-5=(1/3)x-1}}} Multiply



{{{y=(1/3)x-1+5}}} Add 5 to both sides. 



{{{y=(1/3)x+4}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(0,4\right)] is {{{y=(1/3)x+4}}}



 Notice how the graph of {{{y=(1/3)x+4}}} goes through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(0,4\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(1/3)x+4),
 circle(3,5,0.08),
 circle(3,5,0.10),
 circle(3,5,0.12),
 circle(0,4,0.08),
 circle(0,4,0.10),
 circle(0,4,0.12)
 )}}} Graph of {{{y=(1/3)x+4}}} through the points *[Tex \LARGE \left(3,5\right)] and *[Tex \LARGE \left(0,4\right)]