Question

question #11
an equation of a line through (1, - 9) which is perpendicular to the line y = 2x - 5 has what slope and y-intercept?
y - y₁ = m(x - x₁)

Explanation:

Step1: Find the slope of the perpendicular line

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

Step2: Use point - slope form to find the equation of the line

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(1,-9) \) and \( m =-\frac{1}{2} \). Substituting these values, we get \( y-(-9)=-\frac{1}{2}(x - 1) \).
Simplify the equation: \( y + 9=-\frac{1}{2}x+\frac{1}{2} \).
Then, \( y=-\frac{1}{2}x+\frac{1}{2}-9 \).
Calculate \( \frac{1}{2}-9=\frac{1 - 18}{2}=-\frac{17}{2} \). So the equation of the line is \( y =-\frac{1}{2}x-\frac{17}{2} \).

Step3: Identify the slope and y - intercept

For a line in the form \( y = mx + b \), \( m \) is the slope and \( b \) is the y - intercept. From \( y =-\frac{1}{2}x-\frac{17}{2} \), the slope \( m =-\frac{1}{2} \) and the y - intercept \( b=-\frac{17}{2}=- 8.5 \).

Answer:

The slope of the line is \(-\frac{1}{2}\) and the y - intercept is \(-\frac{17}{2}\) (or \(-8.5\)).