line s has an equation of ( y = 8x + 7 ). line t is perpendicular to line s and passes through ( (-8, -2) )…

Explanation:

Step1: Find the slope of line (t)

The slope of line (s) is (m_s = 8). For two perpendicular lines with slopes (m_1) and (m_2), (m_1\times m_2=- 1). Let the slope of line (t) be (m_t). Then (8\times m_t=-1), so (m_t=-\frac{1}{8}).

Step2: Use the point - slope form to find the equation of line (t)

The point - slope form of a line is (y - y_1=m(x - x_1)), where ((x_1,y_1)=(-8,-2)) and (m =-\frac{1}{8}). Substitute the values into the formula: (y-(-2)=-\frac{1}{8}(x - (-8))). Simplify the equation: (y + 2=-\frac{1}{8}(x + 8)).

Step3: Convert to slope - intercept form ((y=mx + b))

Expand the right - hand side: (y+2=-\frac{1}{8}x-1). Subtract 2 from both sides: (y=-\frac{1}{8}x-1 - 2). So (y=-\frac{1}{8}x-3).

Answer:

(y =-\frac{1}{8}x-3)