Solution;

Assume that the number of bikes entering each intersection is equal to the number of the bikes leaving the intersection. For each intersection, this fact can be shown by an equation. 

x₄+120 = x₁ + 250
 
  x₃ + 115 = x₄ + 175
 
  x₂ + 630 = x₃ + 390
 
  x₁ + 70 = x₂ + 120 Rewrite this system of linear equations: $\left\{ \begin{array}{rlrrrlrlrlr}-x_1 & & & & & +& x_4 &=&130 \\& & & & x...... - & x_3 & & &=&-240 \\x_1 &- &x_2 & & & & &=&50 \\\par\end{array}\right.$ The augmented matrix of this system is 
 
 

$A=\left[ \begin{array}{rrrrr}-1& 0& 0 & 1& 130 \\0& 0& 1& -1& 60 \\0& 1& -1& 0& -240 \\1& -1& 0& 0&50\\\end{array} \right]$

The row reduced echelon form of this matrix is
 
 

$A=\left[ \begin{array}{rrrrr}1& 0& 0 & -1& 130 \\0& 1& 0 & -1& -180 \\0& 0& 1& -1& 60 \\0& 0 & 0& 0& 0\\\end{array} \right]$
 
 

Rewrite this as a linear system :

 
  $$\left\{ \begin{array}{rlrrrlrlrlr}x_1 & = & x_4 & + & 130 \......4 & + & -180 \\x_3 & = & x_4 &+ & 60 \\\end{array}\right.$$

Since there is a free variable, this problem has many possible solutions, but x₄ > 180.

As an example if x₄= 400 the solution of the system will be

 
  $$\left\{ \begin{array}{rlrrrlrlrlr}x_1 & =530 \\x_2 & =220 \\x_3 & =460 \\\end{array}\right.$$

1999-11-28