To solve this problem, first, we have to find the slope of the line that has an x-intercept of 3 and a y-intercept of 6. The intercepts can be written as (3,0) and (0,6). Let's use the slope formula
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Where,
[tex]\begin{gathered} x_1=3 \\ x_2=0 \\ y_1=0 \\ y_2=6 \end{gathered}[/tex]Let's use these values to find the slope.
[tex]m=\frac{6-0}{0-3}=\frac{6}{-3}=-2[/tex]The slope of the line that has the given intercepts is m = -2.
Now, we have to find the perpendicular slope of m = -2 using the following rule
[tex]m\cdot m_{\text{perp}}=-1[/tex]Let's replace the slope we found and find the other one.
[tex]m_{\text{perp}}=-\frac{1}{m}=-\frac{1}{-2}=\frac{1}{2}[/tex]The slope of the perpendicular line is 1/2.
Once we have the slope of the new perpendicular line, we use the point-slope formula
[tex]y-y_1=m_{\text{perp}}\cdot(x-x_1)[/tex]Where,
[tex]\begin{gathered} x_1=-6 \\ y_1=4 \\ m_{\text{perp}}=\frac{1}{2} \end{gathered}[/tex]Let's use these values above to find the equation of the new perpendicular line.
[tex]\begin{gathered} y-4=\frac{1}{2}(x-(-6)) \\ y-4=\frac{1}{2}(x+6) \\ y-4=\frac{1}{2}x+\frac{6}{2} \\ y=\frac{1}{2}x+3+4 \\ y=\frac{1}{2}x+7 \end{gathered}[/tex]Therefore, the equation of the new perpendicular line that passes through (-6,4) is
[tex]y=\frac{1}{2}x+7[/tex]The image below shows the graph of this function
