How to Solve the Linear Diophantine Equation With Three Variables

Linear diophantine equations usually appear in the form of two variables. The standard form of a linear diophantine equation is ax + by = c. However, you can easily make the diophantine equation accommodate a third variable by adding it to the left-hand side, yielding ax + by + cz = d. Solving for a three-variable diophantine equation is similar to the process of solving for a two-variable diophantine equation.

Instructions

    • 1

      Identify the variable for which you want to solve. Regardless of the variable you choose, the solution method is similar due to the symmetry involved in the equation.

    • 2

      Subtract the products of the variables not chosen and their coefficients from both sides of the equation. For example, if you want to solve for z, you need to subtract the products ax and by from both sides of the equation. In this example, you would be left with the new equation cz = d - ax - by.

    • 3

      Divide both sides of the equation by the coefficient for the variable that you are solving for. In the example, the coefficient multiplying z is c. Thus, you would divide both sides of the equation by c, yielding z = d/c – (a/c)*x – (b/c)*y. This is the solution to the equation for the variable z.

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