Question

line r has an equation of y + 10 = -2(x - 6). line s is perpendicular to line r and passes through (1, 2). what is the equation of line s?
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 r

The equation of line \( r \) is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope. For \( y + 10=-2(x - 6) \), the slope of line \( r \), \( m_r=-2 \).

Step2: Find slope of line s

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

Step3: Use point - slope form for line s

Line \( s \) passes through the point \( (1,2) \) and has slope \( m_s=\frac{1}{2} \). The point - slope form is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(1,2) \) and \( m = \frac{1}{2} \). So \( y - 2=\frac{1}{2}(x - 1) \).

Step4: Convert to slope - intercept form

Expand the right - hand side: \( y - 2=\frac{1}{2}x-\frac{1}{2} \). Then add 2 to both sides. \( y=\frac{1}{2}x-\frac{1}{2}+2 \). Simplify the constant terms: \( -\frac{1}{2}+2=\frac{-1 + 4}{2}=\frac{3}{2} \). So \( y=\frac{1}{2}x+\frac{3}{2} \).

Answer:

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