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:

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

$l:y=-\dfrac{5}{4}x+b$

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

$0=-\dfrac{5}{4}\cdot0+b\to b=0$

$y=-\dfrac{5}{4}x\ \ \ \ |\cdot4\\\\4y=-5x\ \ \ \ |+5x\\\\5x+4y=0$

Answer: 5x + 4y = 0