# How do you find the equations of two lines with a point of intersection at (2 , -3) and one of the lines passes through the origin of the system of axes?

Feb 5, 2015

We have one line with two points $\left(0 , 0\right) \mathmr{and} \left(2 , - 3\right)$

The equation goes $y = a x + b$

${a}_{1} = \frac{\Delta {y}_{1}}{\Delta {x}_{1}} = \frac{- 3 - 0}{2 - 0} = - \frac{3}{2}$

Fill in what you know to find ${b}_{1}$

${y}_{1} = {a}_{1} x + {b}_{1} \to - 3 = - \frac{3}{2} \cdot 2 + {b}_{1} \to {b}_{1} = 3$

So the equation of the first line is:

${y}_{1} = - \frac{3}{2} x + 3$

The second line needs more data.
If we know it is perpendicular to the first, then

${a}_{2} = - \frac{1}{a} _ 1 = + \frac{2}{3}$

Fill in what you know to find ${b}_{2}$

${y}_{2} = {a}_{2} x + {b}_{2} \to - 3 = \frac{2}{3} \cdot 2 + {b}_{2} \to b = - 4 \frac{1}{3}$

So that equation will be: $\frac{2}{3} x - 4 \frac{1}{3}$