Intersection of right-hand sides

The plan is reported to a reference mark. A right (not vertical) can be defined by an equation: $y = ax + b$

If one considers 2 lines defined by the equations $y = ax + b$ and $y =a' X +b'$ one can already, on the basis of the values of $a, a', b$ and $b'$ , knowledge if there is an intersection according to whether one is in one of the 3 following cases:

If $a=a'$ and $b \ not=b'$ then the lines are parallel and there is no intersection.

If $a=a'$ and $b=b'$ then the 2 lines are confused and there is thus an infinity of points of intersection.

If $a \ not=a'$ , whatever $b$ and $b'$ , there is inevitably a point of intersection. This point is calculated initially in $x$ , which makes it possible to deduce $y$ .

$x = \ frac {(B - b')}{(a' - a)}$ then $y = ax + b$

Demonstration

For the lines of equation there = ax + B and y' = a' X + b' where (a' = 0 and b' = 1), the intersection is for a value of X such as there = y'

ax + B = a' X + b'

ax - a' X = b' - B

(has - a') X = b' - B

x = \ frac {(B - b')}{(a' - a)}

Example

For the lines of equation there = ax + B (where has = 1 and B = 0) and y' = a' X + b' where (a' = 0 and b' = 1), the intersection is for a value of

x = \ frac {(0 - 1)}{(0 - 1)}

and a value of there (and y') of

ax + B 1x + 0 1

The lines are cut for values of X = 1 and there = 1.

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