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.

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 \( (-1 , 1) \).

The results are: the point of intersection of lines \( l_1 \) and \( L_2 \), if any, and whether the three lines are concurrent.

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)
\( L_1: \quad - x - 2y = 3 \)
,
\( L_2: \quad y = -2\)
,
\( L_3: \quad 3x - 4y = 11\)

Cramer's Rule.

Systems of Equations Solver and Calculator.

Equations of Lines in Different Forms.

Online Geometry Calculators and Solvers.