Concurrent Lines Solver and Calculator

Online calculator to find out whether three given lines \( L_1 \), \( L_2 \), and \( L_3 \) are concurrent, meaning they all pass through the same point.

Analytical Solution to Concurrent Lines Problem

Let the three lines be given by the equations:

\[ L_1: \quad a_1 x + b_1 y = c_1 \] \[ L_2: \quad a_2 x + b_2 y = c_2 \] \[ L_3: \quad a_3 x + b_3 y = c_3 \]

First, find the point of intersection of lines \( L_1 \) and \( L_2 \) by solving the system of equations:

\[ \begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases} \]

Using Cramer's rule (determinants), the coordinates of the intersection point are:

\[ x_0 = \frac{ \begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}} = \frac{c_1 b_2 - b_1 c_2}{a_1 b_2 - b_1 a_2} \] \[ y_0 = \frac{ \begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}} = \frac{a_1 c_2 - c_1 a_2}{a_1 b_2 - b_1 a_2} \]

Then, check whether point \( P(x_0, y_0) \) lies on line \( L_3 \) by verifying if:

\[ a_3 x_0 + b_3 y_0 = c_3 \]

📝 Example

Consider the three lines (default values in calculator):

\[ L_1: 2x + y = -1 \] \[ L_2: 3x + 2y = -1 \] \[ L_3: -3x + 4y = 7 \]

Using Cramer's rule:

\[ x_0 = \frac{\begin{vmatrix} -1 & 1 \\ -1 & 2 \end{vmatrix}}{\begin{vmatrix} 2 & 1 \\ 3 & 2 \end{vmatrix}} = \frac{(-1)(2) - (1)(-1)}{(2)(2) - (1)(3)} = \frac{-2 + 1}{4 - 3} = -1 \] \[ y_0 = \frac{\begin{vmatrix} 2 & -1 \\ 3 & -1 \end{vmatrix}}{\begin{vmatrix} 2 & 1 \\ 3 & 2 \end{vmatrix}} = \frac{(2)(-1) - (-1)(3)}{4 - 3} = \frac{-2 + 3}{1} = 1 \]

Check L₃: \( -3(-1) + 4(1) = 3 + 4 = 7 \) ✓

Therefore, the three lines are concurrent at point P(-1, 1).

Use Concurrent Lines Calculator

Enter the coefficients a, b, and c for lines L₁, L₂, and L₃ (as defined above) and press "Calculate".

Three Lines Concurrency Checker

Enter coefficients for lines in the form ax + by = c

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📚 Practice Activities

Check analytically, and using the calculator above, whether these lines are concurrent and find their intersection points:

a) L₁: \(-2x + 7y = 11\)
L₂: \(3x + 7y = 1\)
L₃: \(6x - y = -13\)
b) L₁: \(-7x + y = -32\)
L₂: \(-2x + y = -12\)
L₃: \(x - 7y = 32\)
c) L₁: \(-x - 2y = 3\)
L₂: \(y = -2\) (Note: This is 0·x + 1·y = -2)
L₃: \(3x - 4y = 11\)

⚠️ Special Cases

More References and Links

General Equation of a Line: ax + by = c
Cramer's Rule for Solving Systems
Systems of Equations Solver
Equations of Lines in Different Forms
Online Geometry Calculators and Solvers