Question

write an equation of the line, in slope intercept form, which passes through the given point and is perpendicular to the given line.

1. (-3, 3); $\frac{2y}{2}=\frac{8x}{2}-\frac{6}{2}$
2. (7, 10); $y = \frac{1}{2}x - 9$

Explanation:

Problem 8:

Step1: Find slope of given line

Given line \(2y = 8x - 6\), divide by 2: \(y = 4x - 3\). Slope \(m_1 = 4\). For perpendicular lines, \(m_2=-\frac{1}{m_1}=-\frac{1}{4}\).

Step2: Use point - slope form

Point \((-3,3)\), point - slope formula \(y - y_1=m(x - x_1)\). Substitute: \(y - 3=-\frac{1}{4}(x + 3)\).

Step3: Convert to slope - intercept form

\(y-3 = -\frac{1}{4}x-\frac{3}{4}\), add 3: \(y=-\frac{1}{4}x-\frac{3}{4}+\frac{12}{4}=-\frac{1}{4}x+\frac{9}{4}\).

Step1: Find slope of perpendicular line

Given line \(y=\frac{1}{2}x - 9\), slope \(m_1=\frac{1}{2}\). Perpendicular slope \(m_2=-2\) (since \(m_1\times m_2=-1\)).

Step2: Use point - slope form

Point \((7,10)\), point - slope formula \(y - y_1=m(x - x_1)\). Substitute: \(y - 10=-2(x - 7)\).

Step3: Convert to slope - intercept form

\(y - 10=-2x + 14\), add 10: \(y=-2x+14 + 10=-2x + 24\).

Answer:

\(y = -\frac{1}{4}x+\frac{9}{4}\)

Problem 9: