Question 554308


First let's find the slope of the line through the points *[Tex \LARGE \left(4,-7\right)] and *[Tex \LARGE \left(0,5\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(4,-7\right)]. So this means that {{{x[1]=4}}} and {{{y[1]=-7}}}.

Also, *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point *[Tex \LARGE \left(0,5\right)].  So this means that {{{x[2]=0}}} and {{{y[2]=5}}}.



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



{{{m=(5--7)/(0-4)}}} Plug in {{{y[2]=5}}}, {{{y[1]=-7}}}, {{{x[2]=0}}}, and {{{x[1]=4}}}



{{{m=(12)/(0-4)}}} Subtract {{{-7}}} from {{{5}}} to get {{{12}}}



{{{m=(12)/(-4)}}} Subtract {{{4}}} from {{{0}}} to get {{{-4}}}



{{{m=-3}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(4,-7\right)] and *[Tex \LARGE \left(0,5\right)] is {{{m=-3}}}



Now let's use the point slope formula:



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



{{{y--7=-3(x-4)}}} Plug in {{{m=-3}}}, {{{x[1]=4}}}, and {{{y[1]=-7}}}



{{{y+7=-3(x-4)}}} Rewrite {{{y--7}}} as {{{y+7}}}



{{{y+7=-3x+-3(-4)}}} Distribute



{{{y+7=-3x+12}}} Multiply



{{{y=-3x+12-7}}} Subtract 7 from both sides. 



{{{y=-3x+5}}} Combine like terms. 



{{{y=-3x+5}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(4,-7\right)] and *[Tex \LARGE \left(0,5\right)] is {{{y=-3x+5}}}



 Notice how the graph of {{{y=-3x+5}}} goes through the points *[Tex \LARGE \left(4,-7\right)] and *[Tex \LARGE \left(0,5\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,-3x+5),
 circle(4,-7,0.08),
 circle(4,-7,0.10),
 circle(4,-7,0.12),
 circle(0,5,0.08),
 circle(0,5,0.10),
 circle(0,5,0.12)
 )}}} Graph of {{{y=-3x+5}}} through the points *[Tex \LARGE \left(4,-7\right)] and *[Tex \LARGE \left(0,5\right)]



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