# Concurrent Lines Solver and Calculator

Online calculator to find out whether three given lines $L_1$, $L_2$, and $L_3$ are concurrent, that is they all pass through the same point.

## Analytical Solution to Concurrent Lines Problem

Find, if any, the point of interscetion of three lines.
Let the three lines be given by the equations:
$L_1: \quad a_1 x + b_1 y = c_1$
$L_2: \quad a_2 x + b_2 y = c_2$
$L_3: \quad a_3 x + b_3 y = c_3$

Find the point of intersection, if any, of lines $L_1$ and $L_2$ by solving the
systems of equations corresponding to these two lines.
$\quad a_1 x + b_1 y = c_1$
$\quad a_2 x + b_2 y = c_2$

Using
Cramer's rule (determinants), the $x$ and $y$ coordinates of the point of intersection of lines $L_1$ and $L_2$ are given by:

$x_0 = \dfrac{ \begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}} \quad$ , $\quad y_0 = \dfrac{\begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}}$

We next need to check that the point $x_0, y_0$ is in line $L_3$ by checking that the equation $a_3 x_0 + b_3 y_0 = c_3$ is satisfied.

Example
Let the thee lines given by the equations: (these are the default values shown in the calculator below when you first open the page)
$L_1: \quad 2 x + y = -1$
$L_2: \quad 3 x + 2 y = -1$
$L_1: \quad -3 x + 4 y = 7$

Use Cramer's rule to find the point of intersection of lines $L_1$ and $L_2$

$x_0 = \dfrac{ \begin{vmatrix} -1 & 1\\ -1 & 2 \end{vmatrix} }{\begin{vmatrix} 2 & 1\\ 3 & 2 \end{vmatrix}} = -1 \quad$ , $\quad y_0 = \dfrac{\begin{vmatrix} 2 & -1\\ 3 & -1 \end{vmatrix}}{\begin{vmatrix} 2 & 1\\ 3 & 2 \end{vmatrix}} = 1$

Check whether $L_3$ passes by the point of intersection $(-1 , 1)$ found above by substituting $x$ by $-1$ and $y$ by $1$ in the equation of line $L_3$
$-3 (x_0) + 4 (y_0) = -3 (-1) + 4 (1) = 7$
Hence the right hand and left hand sides of equation $L_3$ are equal and therefore the three line are concurrent at the point $(-1 , 1)$.

## Use Concurrent Lines Calculator and Solver

Enter the coefficients a,b and c as defined above for lines $L_1$, $L_2$ and $L_3$ as real numbers and press "Calculate".
The results are: the point of intersection of lines $l_1$ and $L_2$, if any, and whether the three lines are concurrent.

 Line $L1: \quad$ $a_1$ = 2 , $b_1$ = 1 , $c_1$ = -1 Line $L2: \quad$ $a_2$ = 3 , $b_2$ = 2 , $c_2$ = -1 Line $L3: \quad$ $a_3$ = -3 , $b_3$ = 4 , $c_3$ = 7 Decimal Places = 2

## Activities

Check analytically, and using the above calculator that the following lines are concurrent and find their point of intersections.
a) $L_1: \quad - 2 x + 7y = 11$ ,   $L_2: \quad 3 x + 7 y = 1$ ,   $L_3: \quad 6 x - y = -13$
b) $L_1: \quad -7 x + y = -32$ ,   $L_2: \quad -2 x + y = -12$ ,   $L_3: \quad x - 7y =32$
c) $L_1: \quad - x - 2y = 3$ ,   $L_2: \quad y = -2$ ,   $L_3: \quad 3x - 4y = 11$