Question 555823


First let's find the slope of the line through the points *[Tex \LARGE \left(3,9\right)] and *[Tex \LARGE \left(5,5\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(3,9\right)]. So this means that {{{x[1]=3}}} and {{{y[1]=9}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(5,5\right)].  So this means that {{{x[2]=5}}} and {{{y[2]=5}}}.



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



{{{m=(5-9)/(5-3)}}} Plug in {{{y[2]=5}}}, {{{y[1]=9}}}, {{{x[2]=5}}}, and {{{x[1]=3}}}



{{{m=(-4)/(5-3)}}} Subtract {{{9}}} from {{{5}}} to get {{{-4}}}



{{{m=(-4)/(2)}}} Subtract {{{3}}} from {{{5}}} to get {{{2}}}



{{{m=-2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(3,9\right)] and *[Tex \LARGE \left(5,5\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-9=-2(x-3)}}} Plug in {{{m=-2}}}, {{{x[1]=3}}}, and {{{y[1]=9}}}



{{{y-9=-2x+-2(-3)}}} Distribute



{{{y-9=-2x+6}}} Multiply



{{{y=-2x+6+9}}} Add 9 to both sides. 



{{{y=-2x+15}}} Combine like terms. 



{{{y=-2x+15}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(3,9\right)] and *[Tex \LARGE \left(5,5\right)] is {{{y=-2x+15}}}



 Notice how the graph of {{{y=-2x+15}}} goes through the points *[Tex \LARGE \left(3,9\right)] and *[Tex \LARGE \left(5,5\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-2x+15),
 circle(3,9,0.08),
 circle(3,9,0.10),
 circle(3,9,0.12),
 circle(5,5,0.08),
 circle(5,5,0.10),
 circle(5,5,0.12)
 )}}} Graph of {{{y=-2x+15}}} through the points *[Tex \LARGE \left(3,9\right)] and *[Tex \LARGE \left(5,5\right)]

 

Note: this graph has a slope of -2 and a y-intercept of (0, 15)


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