### Geometry: Lines

### Different descriptions of a line

We have already seen that the equation of a line is uniquely determined by two distinct points on the line. We have also seen that the graph of a *linear function* is a straight line and we described two different ways of writing the *equation of a line*. We will recall these different descriptions and add a third equation for a line.

#y=-{{16}\over{21}}\cdot x-{{32}\over{27}}#

Because the coefficient of #y# in the given equation is not equal to zero, it's posible to rewrite the equation as #y=a\cdot x+b#. We get to this form using reduction:

\[\begin{array}{rcl}

-{{2}\over{7}}\cdot x-{{3}\over{8}}\cdot y&=&{{4}\over{9}}\\&&\phantom{xxx}\blue{\text{the given equation}}\\

-{{3}\over{8}}\cdot y&=&{{2}\over{7}}\cdot x+{{4}\over{9}}\\&&\phantom{xxx}\blue{{{2}\over{7}}\cdot x\text{ added}\text{ on both sides}}\\

y&=&-{{16}\over{21}}\cdot x-{{32}\over{27}}\\&&\phantom{xxx}\blue{\text{left and right hand side divided by } -{{3}\over{8}} \text{, the coeffient of } y}

\end{array}\]

Because the coefficient of #y# in the given equation is not equal to zero, it's posible to rewrite the equation as #y=a\cdot x+b#. We get to this form using reduction:

\[\begin{array}{rcl}

-{{2}\over{7}}\cdot x-{{3}\over{8}}\cdot y&=&{{4}\over{9}}\\&&\phantom{xxx}\blue{\text{the given equation}}\\

-{{3}\over{8}}\cdot y&=&{{2}\over{7}}\cdot x+{{4}\over{9}}\\&&\phantom{xxx}\blue{{{2}\over{7}}\cdot x\text{ added}\text{ on both sides}}\\

y&=&-{{16}\over{21}}\cdot x-{{32}\over{27}}\\&&\phantom{xxx}\blue{\text{left and right hand side divided by } -{{3}\over{8}} \text{, the coeffient of } y}

\end{array}\]

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