what is the equation of the line that is perpendicular to the given line and passes through the point (3…

Explanation:

Step1: Find slope of given line

Use slope - formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ with points $(-3,2)$ and $(0,1)$. So $m_1=\frac{1 - 2}{0+3}=-\frac{1}{3}$.

Step2: Find slope of perpendicular line

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the perpendicular line be $m_2$. Then $m_1\times m_2=-1$. Substituting $m_1 = -\frac{1}{3}$, we get $-\frac{1}{3}m_2=-1$, so $m_2 = 3$.

Step3: Use point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$. Substitute $m = 3$, $x_1 = 3$ and $y_1 = 4$. We have $y - 4=3(x - 3)$.

Step4: Simplify to slope - intercept form

Expand the right - hand side: $y-4 = 3x-9$. Then add 4 to both sides to get $y=3x - 5$.

Answer:

$y = 3x-5$