### 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=-{{1}\over{4}}\cdot x-2#

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}

{{1}\over{2}}\cdot x+2\cdot y&=&-4\\&&\phantom{xxx}\blue{\text{the given equation}}\\

2\cdot y&=&-{{1}\over{2}}\cdot x-4\\&&\phantom{xxx}\blue{{{1}\over{2}}\cdot x\text{ subtracted}\text{ on both sides}}\\

y&=&-{{1}\over{4}}\cdot x-2\\&&\phantom{xxx}\blue{\text{left and right hand side divided by } 2 \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}

{{1}\over{2}}\cdot x+2\cdot y&=&-4\\&&\phantom{xxx}\blue{\text{the given equation}}\\

2\cdot y&=&-{{1}\over{2}}\cdot x-4\\&&\phantom{xxx}\blue{{{1}\over{2}}\cdot x\text{ subtracted}\text{ on both sides}}\\

y&=&-{{1}\over{4}}\cdot x-2\\&&\phantom{xxx}\blue{\text{left and right hand side divided by } 2 \text{, the coeffient of } y}

\end{array}\]

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