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{2}}+{{5}\over{6}}#
Because the coefficient of #y# is not equal to zero in the given equation, it can be reduced to the form #y=a\cdot x+b#. We achieve this form through reduction:
\[\begin{array}{rcl}
-3\cdot x+6\cdot y&=&5 \\&&\phantom{xxx}\blue{\text{the given equation}}\\
6\cdot y&=&3\cdot x+5 \\&&\phantom{xxx}\blue{\text{added }3\cdot x\text{ left and right}}\\
y&=&\displaystyle {{x}\over{2}}+{{5}\over{6}}\\&&\phantom{xxx}\blue{\text{divided left and right by the coefficient of }y}
\end{array}\]
Because the coefficient of #y# is not equal to zero in the given equation, it can be reduced to the form #y=a\cdot x+b#. We achieve this form through reduction:
\[\begin{array}{rcl}
-3\cdot x+6\cdot y&=&5 \\&&\phantom{xxx}\blue{\text{the given equation}}\\
6\cdot y&=&3\cdot x+5 \\&&\phantom{xxx}\blue{\text{added }3\cdot x\text{ left and right}}\\
y&=&\displaystyle {{x}\over{2}}+{{5}\over{6}}\\&&\phantom{xxx}\blue{\text{divided left and right by the coefficient of }y}
\end{array}\]
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