Question

line v has an equation of $y = -\frac{10}{9}x - 3$. perpendicular to line v is line w, which passes through the point (2, 3). what is the equation of line w? 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 w

The slope of perpendicular lines are negative reciprocals. For line \( v \) with slope \( m_v = -\frac{10}{9} \), the slope of line \( w \) (\( m_w \)) is the negative reciprocal: \( m_w=\frac{9}{10} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(2,3) \) and \( m = \frac{9}{10} \). Substitute the values: \( y - 3=\frac{9}{10}(x - 2) \).

Step3: Convert to slope - intercept form

Expand the right - hand side: \( y-3=\frac{9}{10}x-\frac{18}{10} \). Then add 3 to both sides. Since \( 3=\frac{30}{10} \), we have \( y=\frac{9}{10}x-\frac{18}{10}+\frac{30}{10} \). Simplify the constants: \( y=\frac{9}{10}x+\frac{12}{10}=\frac{9}{10}x+\frac{6}{5} \).

Answer:

\( y = \frac{9}{10}x+\frac{6}{5} \)