Question 202967


First let's find the slope of the line through the points *[Tex \LARGE \left(0,5\right)] and *[Tex \LARGE \left(1,1\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(0,5\right)]. So this means that {{{x[1]=0}}} and {{{y[1]=5}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(1,1\right)].  So this means that {{{x[2]=1}}} and {{{y[2]=1}}}.



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



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



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



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



{{{m=-4}}} Reduce



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



Now let's use the point slope formula:



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



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



{{{y-5=-4x+-4(-0)}}} Distribute



{{{y-5=-4x+0}}} Multiply



{{{y=-4x+0+5}}} Add 5 to both sides. 



{{{y=-4x+5}}} Combine like terms. 



{{{y=-4x+5}}} Simplify



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



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

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