Computing the Intersection of Two Lines

Given are usually two linear functions $ff$ and $gg$ with the general equations of the straight line:

• $f(x)=m_1\cdot x+t_{1}f(x)=m\_1\backslash cdot\; x+t\_1$ and

• $g(x)=m_2\cdot x+t_{2}g(x)=m\_2\backslash cdot\; x+t\_2$ .

We are looking for the intersection point $S(a\mathrm{\mid }b)S(a\backslash ,|\backslash ,b)$.

Example

Consider the following linear functions:

• $f(x)=3\cdot x+2f(x)=3\backslash cdot\; x+2$

• $g(x)=2\cdot x+5g(x)=2\backslash cdot\; x+5$.

Calculate the intersection point.

Set the functions equal and bring $xx$ to one side of the equation:

The function value of $ff$ and $gg$ are therefore equal at $x=3x=3$. The point of intersection of the two lines is therefore at $S(3\mathrm{\mid }?)S(3|?)$.

You can now calculate the missing $yy$-value by plugging $x=3x=3$ into $f(x)f(x)$ or $g(x)g(x)$. It doesn't make any difference which of both you choose, since according to the calculation above $f(3)=g(3)f(3)=g(3)$.

$x=3x=3$ plugged into $f(x)f(x)$ yields:

The intersection point is therefore at $S(3\mathrm{\mid }11)S(3|11)$.

General procedure

You set the two functions equal and bring $xx$ to one side of the equation.

The x-value of the intersection point is therefore at $x=\frac{t_2-t_1}{m_1-m_2}x=\backslash frac\{t\_2-t\_1\}\{m\_1-m\_2\}$.

Now plug $x=\frac{t_2-t_1}{m_1-m_2}x=\backslash frac\{t\_2-t\_1\}\{m\_1-m\_2\}$ into $f(x)f(x)$ or $g(x)g(x)$ to get the $yy$-value of the intersection point:

The intersection point of the lines is therefore $S(\frac{t_2-t_1}{m_1-m_2}\mathrm{\mid }{m}_1\cdot \frac{t_2-t_1}{m_1-m_2}+t_1)S(\backslash textcolor\{ff6600\}\{\backslash frac\{t\_2-t\_1\}\{m\_1-m\_2\}\}|\backslash textcolor\{009999\}\{m\_1\backslash cdot\backslash frac\{t\_2-t\_1\}\{m\_1-m\_2\}+t\_1\})$.