Question 536614


First let's find the slope of the line through the points *[Tex \LARGE \left(5,4\right)] and *[Tex \LARGE \left(7,8\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(5,4\right)]. So this means that {{{x[1]=5}}} and {{{y[1]=4}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(7,8\right)].  So this means that {{{x[2]=7}}} and {{{y[2]=8}}}.



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



{{{m=(8-4)/(7-5)}}} Plug in {{{y[2]=8}}}, {{{y[1]=4}}}, {{{x[2]=7}}}, and {{{x[1]=5}}}



{{{m=(4)/(7-5)}}} Subtract {{{4}}} from {{{8}}} to get {{{4}}}



{{{m=(4)/(2)}}} Subtract {{{5}}} from {{{7}}} to get {{{2}}}



{{{m=2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(5,4\right)] and *[Tex \LARGE \left(7,8\right)] is {{{m=2}}}



Now let's use the point slope formula:



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



{{{y-4=2(x-5)}}} Plug in {{{m=2}}}, {{{x[1]=5}}}, and {{{y[1]=4}}}



{{{y-4=2x+2(-5)}}} Distribute



{{{y-4=2x-10}}} Multiply



{{{y=2x-10+4}}} Add 4 to both sides. 



{{{y=2x-6}}} Combine like terms. 



{{{y=2x-6}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(5,4\right)] and *[Tex \LARGE \left(7,8\right)] is {{{y=2x-6}}}



 Notice how the graph of {{{y=2x-6}}} goes through the points *[Tex \LARGE \left(5,4\right)] and *[Tex \LARGE \left(7,8\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,2x-6),
 circle(5,4,0.08),
 circle(5,4,0.10),
 circle(5,4,0.12),
 circle(7,8,0.08),
 circle(7,8,0.10),
 circle(7,8,0.12)
 )}}} Graph of {{{y=2x-6}}} through the points *[Tex \LARGE \left(5,4\right)] and *[Tex \LARGE \left(7,8\right)]



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