Question

Write the equation of the line parallel to 4x-5y=2 and passing through point (0,0)

Answer 1

`k:y=m_1x+b_1;\ l:y=m_2x+b_2\\\\k\ \perp\ l\iff m_1\cdot m_2=-1\\\\k\ ||\ l\iff m_1=m_2`

We have:

`4x-5y=2\ \ \ \ |-4x\\\\-5y=-4x+2\ \ \ \ |:(-5)\\\\y=(4)/(5)x-(2)/(5)\to m_1=(4)/(5)\\\\l:y=m_2x+b\\\\k\ \prep\ l\Rightarrow(4)/(5)m_2=-1\ \ \ \ |\cdot(5)/(4)\\\\m_2=-(5)/(4)`

`l:y=-(5)/(4)x+b`

The line l is passing through point (0, 0). Substitute the coordinates of the point to the equation of a line l:

`0=-(5)/(4)\cdot0+b\to b=0`

`y=-(5)/(4)x\ \ \ \ |\cdot4\\\\4y=-5x\ \ \ \ |+5x\\\\5x+4y=0`

Answer: 5x + 4y = 0