Integer Solutions

The operation Integer on the Solve submenu finds integer solutions to equations and systems of equations.


$\blacktriangleright$ Solve + Integer

3x + 4y = 10, Solution is : $\left\{\vphantom{ x=4N_{1}-10,y=-3N_{1}+10}\right.$x = 4N1 -10, y = - 3N1 + 10$\left.\vphantom{ x=4N_{1}-10,y=-3N_{1}+10}\right\}$

Thus, one solution is given by x = - 10 and y = 10, and every solution is of the form x = 4N1 - 10 and y = - 3N1 + 10, where N1 is any integer. As a check, note that, if x = 4N1 - 10 and y = - 3N1 + 10, then x + 4y = 3(4N1 -10) + 4(- 3N1 + 10) = - 30 + 40 = 10.

In a similar manner, a system of equations can be solved for integer solutions.


$\blacktriangleright$ Solve + Integer

3x + 2y = 5
3x - z = 1
, Solution is : $\left\{\vphantom{ x=5-2N_{1},y=-5+3N_{1},z=-6N_{1}+14}\right.$x = 5 - 2N1, y = - 5 + 3N1, z = - 6N1 + 14$\left.\vphantom{ x=5-2N_{1},y=-5+3N_{1},z=-6N_{1}+14}\right\}$

Indeed, if x, y, and z are given by the stated equations, then, for any integer N1, we have 3x + 2y = 3(5 - 2N1) + 2$\left(\vphantom{ -5+3N_{1}}\right.$ -5 + 3N1$\left.\vphantom{ -5+3N_{1}}\right)$ = 5 and 3x - z = 3$\left(\vphantom{ 5-2N_{1}}\right.$5 - 2N1$\left.\vphantom{ 5-2N_{1}}\right)$ - $\left(\vphantom{ -6N_{1}+14}\right.$ -6N1 + 14$\left.\vphantom{ -6N_{1}+14}\right)$ = 1.