Question 166655
First let's find the slope of the line through the points *[Tex \LARGE \left(\frac{1}{2},3\right)] and *[Tex \LARGE \left(-\frac{1}{2},-2\right)]



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



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



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



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



{{{m=5}}} Reduce



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



Now let's use the point slope formula:



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



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



{{{y-3=5x+5(-1/2)}}} Distribute



{{{y-3=5x-5/2}}} Multiply



{{{y=5x-5/2+3}}} Add 3 to both sides. 



{{{y=5x+1/2}}} Combine like terms. 



So the equation that goes through the points *[Tex \LARGE \left(\frac{1}{2},3\right)] and *[Tex \LARGE \left(-\frac{1}{2},-2\right)] is {{{y=5x+1/2}}}



So the answer in slope intercept form is {{{y=5x+1/2}}} which in standard form is {{{10x-2y=-1}}}