Points in a Box

Related wikis

• Linear Diophantine Equations > Equations with more than 2 Variables

• Extended Euclidean Algorithm

To find the family of solutions, one can use the extended Euclidean algorithm . When solving by hand, I like to follow it in the following manner. First express the variable with the lowest factor:

$x = (1 - 55 y - 77 z) / 35$

We are looking for integral $x$ , so 35 should divide the numerator, or $35 | 1 + 15 y - 7 z$ , or $1 + 15 y - 7 z = 35 k$ for any whole number $k$ . Take first step again, expressing the variable with the lowest factor:

$z = (1 + 15 y - 35 k) / 7$

Again, $z$ should be whole, so $7 | 1 + 15 y - 35 k$ , or $7 | 1 + y$ , or $y = 6 + 7 m$ for any whole number $m$ . So we have $y$ , plugging it above we get $z$ , and plugging both even higher above we get $x$ .

Part two: solving inequalities.