Question 200448


First let's find the slope of the line through the points *[Tex \LARGE \left(-3,7\right)] and *[Tex \LARGE \left(5,-1\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(-3,7\right)] and *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(5,-1\right)].



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



{{{m=(-1-7)/(5--3)}}} Plug in {{{y[2]=-1}}}, {{{y[1]=7}}}, {{{x[2]=5}}}, and {{{x[1]=-3}}}



{{{m=(-8)/(5--3)}}} Subtract {{{7}}} from {{{-1}}} to get {{{-8}}}



{{{m=(-8)/(8)}}} Subtract {{{-3}}} from {{{5}}} to get {{{8}}}



{{{m=-1}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(-3,7\right)] and *[Tex \LARGE \left(5,-1\right)] is {{{m=-1}}}



Now let's use the point slope formula:



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



{{{y-7=-1(x--3)}}} Plug in {{{m=-1}}}, {{{x[1]=-3}}}, and {{{y[1]=7}}}



{{{y-7=-1(x+3)}}} Rewrite {{{x--3}}} as {{{x+3}}}



{{{y-7=-1x+-1(3)}}} Distribute



{{{y-7=-1x-3}}} Multiply



{{{y=-1x-3+7}}} Add 7 to both sides. 



{{{y=-1x+4}}} Combine like terms. 



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



So the equation that goes through the points *[Tex \LARGE \left(-3,7\right)] and *[Tex \LARGE \left(5,-1\right)] is {{{y=-x+4}}}



 Notice how the graph of {{{y=-x+4}}} goes through the points *[Tex \LARGE \left(-3,7\right)] and *[Tex \LARGE \left(5,-1\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-x+4),
 circle(-3,7,0.08),
 circle(-3,7,0.10),
 circle(-3,7,0.12),
 circle(5,-1,0.08),
 circle(5,-1,0.10),
 circle(5,-1,0.12)
 )}}} Graph of {{{y=-x+4}}} through the points *[Tex \LARGE \left(-3,7\right)] and *[Tex \LARGE \left(5,-1\right)]