Question

What is the equation of the line that is perpendicular to y= -3x + 1 and passes through (2,3)?

Answer 1

y = 1/3(x -2) +3

Explanation:

The slope of the given line is the coefficient of x, -3. The slope of the perpendicular line will be the negative reciprocal of that: -1/-3 = 1/3. The line through a point (h, k) with slope m can be written in point-slope form as ...

y = m(x -h) +k

For m=1/3, (h, k) = (2,3), the equation of the line is ...

y = (1/3)(x -2) +3

Answer 2

`\large\boxed{y=(1)/(3)x+2(1)/(3)}`

Explanation:

`\text{Let}\ k:y=_1x+b_1\ \text{and}\ l:y=m_2x+b_2.\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)\\============================\\\\\text{We have}\ y=-3x+1\to m_1=-3.\\\\\text{Therefore}\ m_2=-(1)/(-3)=(1)/(3).\\\\\text{The equation of the searched line:}\ y=(1)/(3)x+b.\\\\\text{The line passes through }(2,\ 3).`

`\text{Put the coordinates of the point to the equation.}\ x=2,\ y=3:\\\\3=(1)/(3)(2)+b\\\\3=(2)/(3)+b\qquad\text{subtract}\ (2)/(3)\ \text{from both sides}\\\\b=2(1)/(3)`