line v has an equation of ( y = -5x + 6 ). line w includes the point ( (-1, -2) ) and is perpendicular to…

Explanation:

Step1: Find the slope of line ( w )

The equation of line ( v ) is ( y=-5x + 6 ), which is in slope - intercept form ( y = mx + b ) (where ( m ) is the slope). So the slope of line ( v), (m_v=-5). If two lines are perpendicular, the product of their slopes is (- 1). Let the slope of line ( w) be (m_w). Then (m_v\times m_w=-1). Substitute (m_v = - 5) into the equation: (-5\times m_w=-1). Solve for (m_w): (m_w=\frac{1}{5}).

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

The point - slope form of a line is (y - y_1=m(x - x_1)), where ((x_1,y_1)=(-1,-2)) and (m = \frac{1}{5}). Substitute the values into the formula: (y-(-2)=\frac{1}{5}(x-(-1))). Simplify the equation: (y + 2=\frac{1}{5}(x + 1)).

Step3: Convert to slope - intercept form

Distribute (\frac{1}{5}) on the right side: (y+2=\frac{1}{5}x+\frac{1}{5}). Subtract (2) from both sides. Since (2=\frac{10}{5}), we have (y=\frac{1}{5}x+\frac{1}{5}-\frac{10}{5}). Combine the constants: (y=\frac{1}{5}x-\frac{9}{5}).

Answer:

(y=\frac{1}{5}x-\frac{9}{5})