Question

find the equation of the line perpendicular to the given line and passing through the given p
y = -2x + 1, (3, 4)
answer

Explanation:

Step1: Identify given line slope

The given line is $y=-2x+1$, so its slope $m_1=-2$.

Step2: Find perpendicular line slope

Perpendicular slopes multiply to $-1$, so $m_2 = \frac{-1}{m_1} = \frac{-1}{-2} = \frac{1}{2}$.

Step3: Use point-slope form

Point-slope formula: $y - y_1 = m_2(x - x_1)$, substitute $(x_1,y_1)=(3,4)$ and $m_2=\frac{1}{2}$.
$y - 4 = \frac{1}{2}(x - 3)$

Step4: Simplify to slope-intercept form

Expand and rearrange:
$y - 4 = \frac{1}{2}x - \frac{3}{2}$
$y = \frac{1}{2}x - \frac{3}{2} + 4$
$y = \frac{1}{2}x + \frac{5}{2}$

Answer:

$y = \frac{1}{2}x + \frac{5}{2}$ (or in standard form: $x - 2y + 5 = 0$)