the equation of line k is y + 10 = 3(x + 3). perpendicular to line k is line l, which passes through the…

Explanation:

Step1: Find the slope of line (k)

The equation of line (k) is (y + 10=3(x + 3)). Rewrite it in slope - intercept form (y=mx + b) (where (m) is the slope). [ \begin{align*} y+10&=3(x + 3)\ y+10&=3x+9\ y&=3x - 1 \end{align*} ] The slope of line (k), (m_{k}=3).

Step2: Find the slope of line (\ell)

If two lines are perpendicular, the product of their slopes is (- 1). Let the slope of line (\ell) be (m_{\ell}). Then (m_{k}\times m_{\ell}=-1). Since (m_{k} = 3), we have (3\times m_{\ell}=-1), so (m_{\ell}=-\frac{1}{3}).

Step3: Use the point - slope form to find the equation of line (\ell)

The point - slope form of a line is (y - y_{1}=m(x - x_{1})) (where ((x_{1},y_{1})) is a point on the line and (m) is the slope). The line (\ell) passes through the point ((5,-5)) and has a slope (m =-\frac{1}{3}). [ \begin{align*} y-(-5)&=-\frac{1}{3}(x - 5)\ y + 5&=-\frac{1}{3}x+\frac{5}{3}\ y&=-\frac{1}{3}x+\frac{5}{3}-5\ y&=-\frac{1}{3}x+\frac{5 - 15}{3}\ y&=-\frac{1}{3}x-\frac{10}{3} \end{align*} ]

Answer:

(y=-\frac{1}{3}x-\frac{10}{3})