The equation of a line

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.

Equation of a line

We can describe the equation of a line in two different ways:

We can describe the equation of a line as \[\blue p \cdot x + \green q \cdot y+\purple r=0\] in which both #p# and #q# can not be #0#.
We can describe the equation of a line as \[\begin{array}{c}y=a \cdot x +b\\ \text{ for an oblique or horizontal line or }\\ x=b\\ \text{ for a vertical line}\end{array}\]

Example

The following equations represent the same line:
\[\blue {-2} \cdot x+\green 3 \cdot y+\purple {-1}=0\] \[y=\frac{2}{3} \cdot x +\frac{1}{3}\]

geogebra plaatje

Reduction

Just like we saw with Solution to a linear equation with two unkowns we can use reduction to rewrite the equations of the form #p \cdot x + q \cdot y+r=0# as #y=a \cdot x+b# or #x=b#.

In the same manner #y=a \cdot x+b# or #x=b# can be rewritten to the form #p \cdot x + q \cdot y+r=0# by means of reduction.

Example

\[\begin{array}{rcl}
-2x+3y-1 &=& 0 \\
3y-1&=&2x \\
3y&=&2x+1 \\
y&=&\frac{2}{3}x+\frac{ 1 }{ 3}
\end{array}\]

If possible, rewrite the equation \[x+6\cdot y=7\] into the form #y=a\cdot x+b# with the correct values of #a# and #b#. If this is not possible, then write #x=b# for the correct value of #b#.

#y=-{{x}\over{6}}+{{7}\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:
\[\begin{array}{rcl}
x+6\cdot y&=&7 \\&&\phantom{xxx}\blue{\text{the given equation}}\\
6\cdot y&=&-x+7\\&&\phantom{xxx}\blue{\text{subtracted }x\text{ left and right}}\\
y&=&\displaystyle -{{x}\over{6}}+{{7}\over{6}}\\&&\phantom{xxx}\blue{\text{left and right divided by the coefficient of }y}
\end{array}\]

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