The original line we are given is:
[tex]y=-3x+6[/tex]This line is in the slope-intercept form:
[tex]y=mx+b[/tex]where m represents the slope, and b represents the y-intercept of the line.
Step 1. Identify the slope of the original line.
By comparing the given line with the slope-intercept form, we see that the slope m is:
[tex]m=-3[/tex]We will rename this slope as m1 because it is the slope of the first line:
[tex]m_1=-3[/tex]Step 2. The second step will be to find the slope of the second line (the perpendicular line). We will call the slope of the second line m2:
[tex]m_2\longrightarrow\text{slope of the perpendicular line}[/tex]And we will need to apply the condition for the slopes of two perpendicular lines:
[tex]m_1\cdot m_2=-1[/tex]Since what we need to find is m2, we solve for it in the previous equation:
[tex]m_2=\frac{-1}{m_1}[/tex]By substituting m1=-2, we can find the slope of the perpendicular line:
[tex]m_2=\frac{-1}{-3}[/tex]The result of this division is:
[tex]m_2=\frac{1}{3}[/tex]Step 3. Once we know the slope of the perpendicular line, we are ready to find the equation that represents it. Remember that we also have a point through which the line passes:
[tex](3,2)[/tex]For reference, we will label the x and y coordinates of this point as follows:
[tex]\begin{gathered} x_0=3 \\ y_0_{}=2 \end{gathered}[/tex]Now, to find the equation of the line we use the point-slope equation:
[tex]y-y_0=m(x-x_0)[/tex]Where x0,y0 represent the point, and m is the slope, in this case, the slope of the perpendicular line:
[tex]y-y_0=m_2(x-x_0)[/tex]We substitute m2, x0, and y0:
[tex]y-2=\frac{1}{3}(x-3)[/tex]And simplify the result in order to solve for y:
[tex]\begin{gathered} y-2=\frac{1}{3}x-\frac{1}{3}\cdot3 \\ y-2=\frac{1}{3}x-1 \\ y=\frac{1}{3}x-1+2 \\ y=\frac{1}{3}x+1 \end{gathered}[/tex]And we have found the equation that represents the perpendicular line.
Answer:
[tex]y=\frac{1}{3}x+1[/tex]