Definition. A line pencil is the set of straight line passing through a given point.

Arbitrarily varying the coefficient k in the equation of a line passing through a point М₀(х₀,у₀), we obtain equations of all straight lines passing through the point with coordinates х₀, у₀.

All these straight lines constitute the line pencil centered at М₀.

A line pencil can also be specified by two equations of lines from this pencil (which determine of these lines) the common point М₀(х₀, у₀) by the coordinates of its center.

Suppose given intersecting straight lines with general equations

,

.

Consider the equations

, (13)

where q₁ and q₂ undetermined multipliers, not both zero.

Let us prove that, for any q₁ and q₂, the straight line (13) passes through the point М₀ (х₀; у₀), which is the intersection point of the two given lines.

The coordinates of the point М₀ satisfy both equations

,

Consequently, the coordinates of М₀satisfy equation (13), i.e.,

.

Thus, the point М₀ belongs to the straight line (13) for any q₁ and q₂; but for particular values of q₁ and q₂, we obtain some straight line from the pencil.

In practice, it is more convenient to write an equation of a pencil with one parameter l:

.

Example. Draw a straight line through the intersection point of the lines х+у–3=0 and 2х–у+5=0 so that it passes through the point М(3;5).

Let us write the equation of the pencil of lines according to the formula. To find the required line, we must find the value of the parameter l:

We substitute the coordinates of the point М(3;5) into the equation of the pencil:

,

.

Substituting l in to the equation of the pencil, we obtain the required equation:

;

.