Alg6.1

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For these pairs of points, find the midpoint, distance, slope, and equation of the line.

(-2,10),(4,9)\,

To find the midpoint, average the x coordinates and y coordinates. The midpoint is

\left({\frac  {-2+4}{2}},{\frac  {10+9}{2}}\right)=\left({\frac  {2}{2}},{\frac  {19}{2}}\right)=\left(1,{\frac  {19}{2}}\right)\,

To find the (always zero or positive) distance, use the formula d=+{\sqrt  {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\,

d={\sqrt  {(-2-4)^{2}+(10-9)^{2}}}={\sqrt  {(-6)^{2}+1^{2}}}={\sqrt  {36+1}}={\sqrt  {37}}\,

To find the slope, use the formula m={\frac  {y_{2}-y_{1}}{x_{2}-x_{1}}}\,

m={\frac  {9-10}{4-(-2)}}={\frac  {-1}{6}}\,

The equations of the line are

Form 1: y=mx+b\,

Plug in one known point (say, (4,9)\,) and the calculated slope.

9={\frac  {-1}{6}}\cdot 4+b\,

b=9+{\frac  {2}{3}}={\frac  {29}{3}}\,

Now plug b and m into the line equation:

  • y={\frac  {-1}{6}}x+{\frac  {29}{3}}\,

Form 2: (y-y_{1})=m(x-x_{1})\,

Plug in one known point (say, (-2,10)\,) and the calculated slope.

(y-10)={\frac  {-1}{6}}(x-(-2))\,

y={\frac  {-1}{6}}x-{\frac  {2}{6}}+10={\frac  {-1}{6}}x+{\frac  {58}{6}}\,

  • y={\frac  {-1}{6}}x+{\frac  {29}{3}}\,

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