# Point of Intersection of Two Lines

The point of intersection of two lines ${a_1}x + {b_1}y + {c_1} = 0$ and ${a_2}x + {b_2}y + {c_2} = 0$ is given by

where ${a_1}{b_2} - {a_2}{b_1} \ne 0$.

To prove this formula we have the given equations of straight lines:

We solve the above equations using the simultaneous method.

Multiplying equation (i) by ${b_2}$, we have

Multiplying equation (ii) by ${b_1}$, we have

Now subtracting (iv) from equation (iii), we get

Multiplying equation (i) by ${a_2}$, we have

Multiplying equation (ii) by ${a_1}$, we have

Now subtracting (vi) from equation (v), we get

This shows that the point of intersection is

Note: If ${a_1}{b_2} - {a_2}{b_1} = 0$, then lines (i) and (ii) will have no common point and therefore, these will be parallel lines.

Now

This is the condition for lines (i) and (ii) to be parallel lines.