How do you write the equation $y-2=3\left( x-1 \right)$ in the slope-intercept form ?

Hint: We recall the three forms of writing a linear equation which are the general form$Ax+By+C=0$, the slope intercept form $y=mx+c$ and the standard form$Ax+By=C$. We add 2 both sides of the given equation and simplify the right hand side to convert the given equation into slope point form.

Complete step by step solution:
We know from the Cartesian coordinate system that every linear equation $Ax+By+C=0$can be represented as a line. If the line is inclined with positive $x-$axis at an angle $\theta $ then its slope is given by $m=\tan \theta $ and if it cuts $y-$axis at a point $\left( 0,c \right)$ from the origin the $y-$intercept is given by $c$. The slope-intercept form of equation is given by
\[y=mx+c....\left( 1 \right)\]
We know that the standard form of linear equation otherwise also known as intercept form is written with constant $C$on the right side of equality sign as
\[Ax+By=C...\left( 2 \right)\]
We are given in the following equation in the question
\[y-2=3\left( x-1 \right)\]
We add 2 both sides of the given equation to have
\[\Rightarrow y=3\left( x-1 \right)+2\]
We simplify the right hand side of the above equation to have;
\[\begin{align}
  & \Rightarrow y=3x-3+2 \\
 & \Rightarrow y=3x-1 \\
\end{align}\]
We see that the above equation is in slope-point form. We compare it with general slope point equation $y=mx+c$ to have the slope and intercept as
\[\begin{align}
  & m=3 \\
 & c=-1 \\
\end{align}\]

Note: We note that the given equation is in the Cartesian slope point equation. If $\left( {{x}_{1}},{{y}_{1}} \right)$ is a point on the line with slope $m$ then the equation is given by $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$. Here the given equation $y-2=3\left( x-1 \right)$ has slope 3 and passes through $\left( 2,1 \right)$. The other Cartesian equation of line is $y-{{y}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\left( x-{{x}_{1}} \right)$ where $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are two points on the line.