Question

Line L has equation 2x - 3y = 5.

Line M passes through the point (3, -10) and is parallel to line L.
Determine the equation for line M.

Answer 1

`\text{Let}\ k:y=m_1x+b_1\ \text{and}\ l:y=m_2x+b_2,\ \text{then}\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)\\\\l\ \parallel\ k\iff m_1=m_2\\\\\text{Convert the equation to the slope-intercept form}\ y=mx+b.\\\\l:2x-3y=5\qquad\text{subtract 2x from both sides}\\\\-3y=-2x+5\qquad\text{divide both sides by (-3)}\\\\l:y=(2)/(3)x-(5)/(3)\to m_1=(2)/(3)\\\\m_2=(2)/(3)\\\\\text{Therefore we have the equation of a line}\ m:y=-(3)/(2)x+b.`

`\text{We know the line passes through the point (3, -10)}.\\\text{Substitute the coordinates of the point to the equation of a line:}\\\\-10=(2)/(3)(3)+b\\\\-10=2+b\qquad\text{substract 2 from both sides}\\\\-12=b\to b=-12\\\\y=(2)/(3)x-12\qquad\text{multiply both sides by 3}\\\\3y=2x-12\qquad\text{subtract 2x from both sides}\\\\-2x+3y=-12\qquad\text{change the signs}\\\\2x-3y=12\\\\Answer:\ \boxed{m:y=(2)/(3)x-12}\to\boxed{m:2x-3y=12}`

Answer 2

Answer: 2x - 3y = 36

Step-by-step explanation: is correct. Line L has slope 2/3, use this and the given point to apply the point-slope formula, y - y1 = m(x - x1).