Question 160008

First let's find the slope of the line through the points *[Tex \LARGE \left(-1,-1\right)] and *[Tex \LARGE \left(4,4\right)]



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



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



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



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



{{{m=1}}} Reduce



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



{{{y--1=1(x+1)}}} Rewrite {{{x--1}}} as {{{x+1}}}



{{{y+1=1(x+1)}}} Rewrite {{{y--1}}} as {{{y+1}}}



{{{y+1=1x+1(1)}}} Distribute



{{{y+1=1x+1}}} Multiply



{{{y=1x+1-1}}} Subtract 1 from both sides. 



{{{y=1x+0}}} Combine like terms. 



{{{y=1x}}} Remove the trailing zero



{{{y=x}}} Simplify



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



 Notice how the graph of {{{y=x}}} goes through the points *[Tex \LARGE \left(-1,-1\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,x),
 circle(-1,-1,0.08),
 circle(-1,-1,0.10),
 circle(-1,-1,0.12),
 circle(4,4,0.08),
 circle(4,4,0.10),
 circle(4,4,0.12)
 )}}} Graph of {{{y=x}}} through the points *[Tex \LARGE \left(-1,-1\right)] and *[Tex \LARGE \left(4,4\right)]