Question

line pl is represented by the equation $y = 2x + 1$. determine the equation, in slope - intercept form, of the line wq that is perpendicular to line pl and passes through the point $w(0, -7)$. $y = \square$

Explanation:

Step1: Find slope of PL

The equation of line \( PL \) is \( y = 2x + 1 \), which is in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept). So, the slope of line \( PL \), \( m_1=2 \).

Step2: Find slope of WQ

If two lines are perpendicular, the product of their slopes is \( - 1 \). Let the slope of line \( WQ \) be \( m_2 \). Then \( m_1\times m_2=-1 \). Substituting \( m_1 = 2 \), we get \( 2\times m_2=-1 \). Solving for \( m_2 \), we have \( m_2=-\frac{1}{2} \).

Step3: Use point - slope form to find equation of WQ

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope of the line. The line \( WQ \) passes through the point \( W(0,-7) \), so \( x_1 = 0 \), \( y_1=-7 \) and \( m =-\frac{1}{2} \).
Substituting these values into the point - slope form:
\( y-(-7)=-\frac{1}{2}(x - 0) \)
Simplify the left - hand side: \( y + 7=-\frac{1}{2}x \)
Then, subtract 7 from both sides to get the slope - intercept form (\( y=mx + b \)): \( y=-\frac{1}{2}x-7 \)

Answer:

\( y = -\frac{1}{2}x - 7 \)