# 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.

## Results

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

## More References and Links

General Equation of a Line: ax + by = c .Cramer's Rule .

Systems of Equations Solver and Calculator .

Equations of Lines in Different Forms .

Online Geometry Calculators and Solvers .