Question 274981


First let's find the slope of the line through the points *[Tex \LARGE \left(4,1\right)] and *[Tex \LARGE \left(0,-4\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(4,1\right)]. So this means that {{{x[1]=4}}} and {{{y[1]=1}}}.

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



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



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



{{{m=5/4}}} Reduce



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



Now let's use the point slope formula:



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



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



{{{y-1=(5/4)x+(5/4)(-4)}}} Distribute



{{{y-1=(5/4)x-5}}} Multiply



{{{y=(5/4)x-5+1}}} Add 1 to both sides. 



{{{y=(5/4)x-4}}} Combine like terms. 



{{{y=(5/4)x-4}}} Simplify



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



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