Question 153129


First let's find the slope of the line through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(3,5\right)]



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



{{{m=(5-4)/(3-2)}}} Plug in {{{y[2]=5}}}, {{{y[1]=4}}}, {{{x[2]=3}}}, {{{x[1]=2}}}, , 



{{{m=(1)/(3-2)}}} Subtract {{{4}}} from {{{5}}} to get {{{1}}}



{{{m=(1)/(1)}}} Subtract {{{2}}} from {{{3}}} to get {{{1}}}



{{{m=1}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(3,5\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-2)}}} Plug in {{{m=1}}}, {{{x[1]=2}}}, and {{{y[1]=4}}}



{{{y-4=1x+1(-2)}}} Distribute



{{{y-4=1x-2}}} Multiply



{{{y=1x-2+4}}} Add 4 to both sides. 



{{{y=1x+2}}} Combine like terms. 



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



So the equation that goes through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(3,5\right)] is {{{y=x+2}}}



 Notice how the graph of {{{y=x+2}}} goes through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(3,5\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,x+2),
 circle(2,4,0.08),
 circle(2,4,0.10),
 circle(2,4,0.12),
 circle(3,5,0.08),
 circle(3,5,0.10),
 circle(3,5,0.12)
 )}}} Graph of {{{y=x+2}}} through the points *[Tex \LARGE \left(2,4\right)] and *[Tex \LARGE \left(3,5\right)]