Question

line v has an equation of 2x - y = 5. line w is perpendicular to line v and passes through (8, -2). 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 v

Rewrite \(2x - y = 5\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope).
We get \(y = 2x-5\). So the slope of line \(v\), \(m_v=2\).

Step2: Find slope of line w

If two lines are perpendicular, the product of their slopes is \(- 1\). Let the slope of line \(w\) be \(m_w\).
We have \(m_v\times m_w=-1\). Substituting \(m_v = 2\), we get \(2\times m_w=-1\), so \(m_w=-\frac{1}{2}\).

Step3: Use point - slope form to find equation of line w

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(8,-2)\) and \(m = m_w=-\frac{1}{2}\).
Substitute the values: \(y-(-2)=-\frac{1}{2}(x - 8)\).
Simplify: \(y + 2=-\frac{1}{2}x+4\).

Step4: Rewrite in slope - intercept form

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

Answer:

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