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 $3\cdot x+3\cdot y=4$ 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 $b$ .

$y=-x+{{4}\over{3}}$

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:
$\begin{array}{rcl} 3\cdot x+3\cdot y&=&4 \\&&\phantom{xxx}\blue{\text{the given equation}}\\ 3\cdot y&=&-3\cdot x+4\\&&\phantom{xxx}\blue{\text{subtracted }3\cdot x\text{ left and right}}\\ y&=&\displaystyle -x+{{4}\over{3}}\\&&\phantom{xxx}\blue{\text{divided left and right by the coefficient of }y} \end{array}$

New example