Change $3x + 6y = 12$ into slope intercept form.

Hint: Slope intercept form: $y = mx + b,{\text{m is the slope of the line}}$
So here we have to convert the given equation in the form of the above equation $y = mx + b$. So we need to manipulate the given equation in such a way that we can perform any arithmetic operations on both LHS and RHS equally at the same time such that the equality of the given equation doesn’t change.

Complete step by step solution:
Given
$3x + 6y = 12.........................\left( i \right)$
We need to change (i) to slope intercept form which is:
$y = mx + b............................\left( {ii} \right)$
So to convert the given equation (i) first we have to subtract the term $3x$ to both LHS and RHS.
Such that:
$
\Rightarrow 3x - 3x + 6y = 12 - 3x \\
\Rightarrow 6y = 12 - 3x................\left( {iii} \right) \\
$
Now to convert and express the equation (iii) in the form of (i) we need to isolate the term $y$ from the equation (iii).
For that we need to divide the equation (iii) with 6 from both LHS and RHS.
Such that:
$
\Rightarrow \dfrac{{6y}}{6} = \dfrac{{12 - 3x}}{6} \\
\Rightarrow y = \dfrac{{12}}{6} - \dfrac{{3x}}{6}.....................\left( {iv} \right) \\
$
Now on comparing (i) and (iv) we can say that we can get the form by simply simplifying the equation
(iv) such that:
$
\Rightarrow y = - \dfrac{{3x}}{6} + \dfrac{{12}}{6} \\
\Rightarrow y = - \dfrac{x}{2} + 2.....................\left( v \right) \\

$
Such that now equation (v) and (iv) are similar now. Also on comparing we can say that $m = -
\dfrac{1}{2}$.
Therefore the slope intercept form of $3x + 6y = 12$ is $y = - \dfrac{x}{2} + 2$.

Note:
We also have an equation $\dfrac{x}{a} + \dfrac{y}{b} = 1$ where $a,b$ are the x intercept and y intercept respectively. By using this equation we can find the x intercept and y intercept. Also while converting an equation as above one must keep in mind that the arithmetic calculations are done correctly since chances of random errors are higher in these types of questions.