### Systems of linear equations: An equation of a line

### The equation of a line

We have *seen* that solutions of the form #\blue p \cdot x + \green q\cdot y+\purple r=0# have a line as solutions. We have also *seen* that the linear formula #y = a\cdot x+b# has a line as a graph. Hence, there are two ways of describing the equation of a line.

#y=-{{x}\over{6}}+{{5}\over{6}}#

Because the coefficient of #y# is not equal to zero in the given equation, it is possible to reduce the equation to the form #y=a\cdot x+b#. We achieve this form through reduction:

\[\begin{array}{rcl}

-x-6\cdot y&=&-5 \\&&\phantom{xxx}\blue{\text{the given equation}}\\

-6\cdot y&=&x-5 \\&&\phantom{xxx}\blue{\text{added }x\text{ left and right}}\\

y&=&\displaystyle -{{x}\over{6}}+{{5}\over{6}}\\&&\phantom{xxx}\blue{\text{left and right divided by the coefficient of }y}

\end{array}\]

Because the coefficient of #y# is not equal to zero in the given equation, it is possible to reduce the equation to the form #y=a\cdot x+b#. We achieve this form through reduction:

\[\begin{array}{rcl}

-x-6\cdot y&=&-5 \\&&\phantom{xxx}\blue{\text{the given equation}}\\

-6\cdot y&=&x-5 \\&&\phantom{xxx}\blue{\text{added }x\text{ left and right}}\\

y&=&\displaystyle -{{x}\over{6}}+{{5}\over{6}}\\&&\phantom{xxx}\blue{\text{left and right divided by the coefficient of }y}

\end{array}\]

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