Question 635639


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



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

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(-3,7\right)].  So this means that {{{x[2]=-3}}} and {{{y[2]=7}}}.



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



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



{{{m=(4)/(-3-1)}}} Subtract {{{3}}} from {{{7}}} to get {{{4}}}



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



{{{m=-1}}} Reduce



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



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



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



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



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



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



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



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

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