Question

what is the equation in slope-intercept form of the line that passes through the point (2, -2) and is perpendicular to the line represented by $y = \frac{2}{5}x + 2$?

a $y = \frac{5}{2}x - 7$

b $y = \frac{5}{2}x + 7$

c $y = -\frac{5}{2}x - 3$

d $y = -\frac{5}{2}x + 3$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of the given line \( y = \frac{2}{5}x + 2 \) is \( m_1=\frac{2}{5} \). For two perpendicular lines, the product of their slopes is \( - 1 \), i.e., \( m_1\times m_2=-1 \). So, \( \frac{2}{5}\times m_2=-1 \), solving for \( m_2 \) gives \( m_2 =-\frac{5}{2} \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(2,-2) \) and \( m =-\frac{5}{2} \). Substituting these values, we get \( y-(-2)=-\frac{5}{2}(x - 2) \).
Simplify the left - hand side: \( y + 2=-\frac{5}{2}(x - 2) \).
Expand the right - hand side: \( y+2=-\frac{5}{2}x + 5 \).
Subtract 2 from both sides: \( y=-\frac{5}{2}x+5 - 2 \), which simplifies to \( y =-\frac{5}{2}x + 3 \).

Answer:

D. \( y =-\frac{5}{2}x + 3 \)