Write an equation in slope-intercept form for the line that satisfies each set of conditions.

Passes through $\left(3,2\right)$, perpendicular to the graph of $4x-3y=12$

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

The y-intercept of a graph is when the x-coordinate is 0. The point where the graph of the equation cuts the y-axis.

It is provided that line passes through the point $\left(3,2\right)$and perpendicular to the graph of $4x-3y=12$.

Product of slopes of perpendicular lines is $-1$. This means slope of line is $-1$divided by slope of the line $4x-3y=12$.

Rewrite the equation in slope-intercept form to find the slope of the line.

Subtract both sides by 4x.

$\begin{array}{c}4x-3y-4x=12-4x\\ -3y=12-4x\end{array}$

Divide both sides by –3.

$\begin{array}{c}y=-\frac{12}{3}+\frac{4x}{3}\\ y=\frac{4}{3}x-4\end{array}$

Therefore, the slope of the line is $m=\frac{4}{3}$.

Now that line passes through the point $\left(3,2\right)$with slope localid="1647619949487" $m=-\frac{3}{4}$.

So, to find the y-intercept that is value of c, follow the step provided below.

Substitute x as 3, y as 2 and m as localid="1647619967984" $-\frac{3}{4}$in the equation $y=mx+c$.

localid="1647620044108" $\begin{array}{c}2=-\frac{3}{4}\left(3\right)+c\\ 2=-\frac{9}{4}+c\\ 2+\frac{9}{4}=c\\ c=\frac{17}{4}\end{array}$

Therefore, y-intercept is localid="1647620059927" $\frac{17}{4}$.

Recall the equation of line in slope-intercept form that is $y=mx+c$.

Where m is the slope and c is the intercept of y-axis.

Replace m by localid="1647620076185" $-\frac{3}{4}$ and c by localid="1647620088232" $\frac{17}{4}$to obtain equation of line in slope-intercept form.

localid="1647620114158" $y=-\frac{3}{4}x+\frac{17}{4}$

Hence, equation of line passing through the point $\left(3,2\right)$and perpendicular to the graph of $4x-3y=12$is localid="1647620124546" $y=-\frac{3}{4}x+\frac{17}{4}$.