Analytics geometry: point of lines intersection calculator
Calculator shows how your straight lines pair are positioned on one plane. It draws a chart of given straight lines if possible. Also, it calculates an angle between those lines.

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Calculations data - line scope and free coefficient of your lines

The first line
First line scope (a1)
Free parameter of the first line (b1)
The second line
Second line scope (a2)
Free parameter of the second line (b2)
Others
Prefered angle measure unit

Results - what we can say about your lines

Summary
Formula of the first functionShow sourcey=2 x+3y=2~ x+3
Formula of the second functionShow sourcey=4 x+5y=-4~ x+5
Lines cross point
Point of intersection of linesShow source(13,113)\left(\frac{1}{3},\frac{11}{3}\right)
Lines position
Are lines parallelno
Are lines perpendicularno
Angle between lines
Tangent of angleShow source67\frac{6}{7}
AngleShow source0.7086262721276703[rad]0.7086262721276703 \left[rad\right]

Functions graph

Some facts

  • Two lines k and l are parallel if they have identical line scope:
    k:y=a x+b1l:y=a x+b2\begin{aligned} k: y & = \boxed {a} ~ x + b_1 \\ l: y & = \boxed {a} ~ x + b_2 \end{aligned}
    where:
    • a - line scope, the same for both straight lines,
    • b1 - free coefficient of the first straight line,
    • b2 - free coefficient of the second straight line.
  • Two lines k and l are perpendicular if line scope of the first is negative inverse of the second one:
    a1=1a2a_1 = - \frac{1}{a_2}
    k:y=a x+b1l:y=1a x+b2\begin{aligned} k: y & = \boxed{a} ~ x + b_1 \\ l: y & = \boxed{- \frac{1}{a}} ~ x + b_2 \end{aligned}
    where:
    • a - scope of the first line,
    • b1 - free coefficient of the first straight line,
    • b2 - free coefficient of the second straight line.
  • If lines are parallel, then they have no common point. Colloquially, one never crosses another one.
  • If lines are not parallel, then they intersect exactly once in the point with below coordinates:
    {Px=b2b1a1a2Py=a1 b2b1a1a2+b1\begin{cases} P_x = \frac {b_2 - b_1}{a_1 - a_2} \\ P_y = a_1 ~ \frac{b_2 - b_1}{a_1 - a_2} + b_1 \end{cases}
    where:
    • a1 - scope of the first line,
    • a2 - scope of the second line,
    • b1 - free coefficient of the first straight line,
    • b2 - free coefficient of the second straight line.
  • To calculate the measure of angle created between straight lines we can use the following formula:
    α=arctan(a2a11+a1 a2)\alpha = \arctan\left(\left| \frac{a_2 - a_1}{1 + a1 ~a2} \right| \right)
    where:
    • α\alpha - angle measure between lines,
    • a1a_1 - scope of the first line,
    • a2a_2 - scope of the second line.
  • If you want to know more about straight line, its formula and properties, check our other calculator: Straight line.

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