Question

line u has an equation of $y + 6 = \frac{3}{2}(x - 6)$. line v is perpendicular to line u and passes through $(2, -1)$. what is the equation of line v?
write the equation in slope-intercept form. write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Explanation:

Step1: Find slope of line u

The equation of line \( u \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope. For \( y + 6=\frac{3}{2}(x - 6) \), the slope of line \( u \), \( m_u=\frac{3}{2} \).

Step2: Find slope of line v

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of line \( v \) be \( m_v \). Then \( m_u\times m_v=- 1 \). Substituting \( m_u = \frac{3}{2} \), we get \( \frac{3}{2}\times m_v=-1 \). Solving for \( m_v \), we have \( m_v=-\frac{2}{3} \).

Step3: Use point - slope form for line v

Line \( v \) passes through the point \( (2,-1) \) and has slope \( m_v = -\frac{2}{3} \). The point - slope form is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(2,-1) \) and \( m = -\frac{2}{3} \). So \( y-(-1)=-\frac{2}{3}(x - 2) \), which simplifies to \( y + 1=-\frac{2}{3}x+\frac{4}{3} \).

Step4: Convert to slope - intercept form

Subtract 1 from both sides. \( y=-\frac{2}{3}x+\frac{4}{3}-1 \). Since \( 1=\frac{3}{3} \), then \( y=-\frac{2}{3}x+\frac{4 - 3}{3}=-\frac{2}{3}x+\frac{1}{3} \).

Answer:

\( y = -\frac{2}{3}x+\frac{1}{3} \)