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={{2\cdot x}\over{3}}-2#
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:
\[\begin{array}{rcl}
2\cdot x-3\cdot y&=&6 \\&&\phantom{xxx}\blue{\text{the given equation}}\\
-3\cdot y&=&-2\cdot x+6\\&&\phantom{xxx}\blue{\text{subtracted }2\cdot x\text{ left and right}}\\
y&=&\displaystyle {{2\cdot x}\over{3}}-2\\&&\phantom{xxx}\blue{\text{left and right divided by the coefficient of }y}
\end{array}\]
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:
\[\begin{array}{rcl}
2\cdot x-3\cdot y&=&6 \\&&\phantom{xxx}\blue{\text{the given equation}}\\
-3\cdot y&=&-2\cdot x+6\\&&\phantom{xxx}\blue{\text{subtracted }2\cdot x\text{ left and right}}\\
y&=&\displaystyle {{2\cdot x}\over{3}}-2\\&&\phantom{xxx}\blue{\text{left and right divided by the coefficient of }y}
\end{array}\]
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