Question 205949


First let's find the slope of the line through the points *[Tex \LARGE \left(0,2\right)] and *[Tex \LARGE \left(4,4\right)]



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point *[Tex \LARGE \left(0,2\right)]. So this means that {{{x[1]=0}}} and {{{y[1]=2}}}.

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



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



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



{{{m=(2)/(4-0)}}} Subtract {{{2}}} from {{{4}}} to get {{{2}}}



{{{m=(2)/(4)}}} Subtract {{{0}}} from {{{4}}} to get {{{4}}}



{{{m=1/2}}} Reduce



So the slope of the line that goes through the points *[Tex \LARGE \left(0,2\right)] and *[Tex \LARGE \left(4,4\right)] is {{{m=1/2}}}



Now let's use the point slope formula:



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



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



{{{y-2=(1/2)x+(1/2)(-0)}}} Distribute



{{{y-2=(1/2)x+0}}} Multiply



{{{y=(1/2)x+0+2}}} Add 2 to both sides. 



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



{{{y=(1/2)x+2}}} Simplify



So the equation that goes through the points *[Tex \LARGE \left(0,2\right)] and *[Tex \LARGE \left(4,4\right)] is {{{y=(1/2)x+2}}}



 Notice how the graph of {{{y=(1/2)x+2}}} goes through the points *[Tex \LARGE \left(0,2\right)] and *[Tex \LARGE \left(4,4\right)]. So this visually verifies our answer.

 {{{drawing( 500, 500, -10, 10, -10, 10,
 graph( 500, 500, -10, 10, -10, 10,(1/2)x+2),
 circle(0,2,0.08),
 circle(0,2,0.10),
 circle(0,2,0.12),
 circle(4,4,0.08),
 circle(4,4,0.10),
 circle(4,4,0.12)
 )}}} Graph of {{{y=(1/2)x+2}}} through the points *[Tex \LARGE \left(0,2\right)] and *[Tex \LARGE \left(4,4\right)]