Question 382114


First let's find the slope of the line through the points *[Tex \LARGE \left(6,4\right)] and *[Tex \LARGE \left(4,6\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(6,4\right)]. So this means that {{{x[1]=6}}} and {{{y[1]=4}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(4,6\right)].  So this means that {{{x[2]=4}}} and {{{y[2]=6}}}.



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



{{{m=(6-4)/(4-6)}}} Plug in {{{y[2]=6}}}, {{{y[1]=4}}}, {{{x[2]=4}}}, and {{{x[1]=6}}}



{{{m=(2)/(4-6)}}} Subtract {{{4}}} from {{{6}}} to get {{{2}}}



{{{m=(2)/(-2)}}} Subtract {{{6}}} from {{{4}}} to get {{{-2}}}



{{{m=-1}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(6,4\right)] and *[Tex \LARGE \left(4,6\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-4=-1(x-6)}}} Plug in {{{m=-1}}}, {{{x[1]=6}}}, and {{{y[1]=4}}}



{{{y-4=-1x+-1(-6)}}} Distribute



{{{y-4=-1x+6}}} Multiply



{{{y=-1x+6+4}}} Add 4 to both sides. 



{{{y=-1x+10}}} Combine like terms. 



{{{y=-x+10}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(6,4\right)] and *[Tex \LARGE \left(4,6\right)] is {{{y=-x+10}}}



 Notice how the graph of {{{y=-x+10}}} goes through the points *[Tex \LARGE \left(6,4\right)] and *[Tex \LARGE \left(4,6\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-x+10),
 circle(6,4,0.08),
 circle(6,4,0.10),
 circle(6,4,0.12),
 circle(4,6,0.08),
 circle(4,6,0.10),
 circle(4,6,0.12)
 )}}} Graph of {{{y=-x+10}}} through the points *[Tex \LARGE \left(6,4\right)] and *[Tex \LARGE \left(4,6\right)]

 


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