Solve Linear Diophantine Equations in 4 Simple Steps [Guide]

Need an integer‑only solution fast? This guide shows exactly how to solve linear diophantine equations without endless trial‑and‑error. Follow the four‑step workflow, pick the right t, and you’ll have a valid integer pair in seconds.

Why the GCD Test Is Your First Gatekeeper

The hidden rule is simple: the gcd condition must hold.
Take the coefficients a and b of ax + by = c. Compute

g = gcd(a, b)

If g does not divide c, no integer solution exists—stop right there. This single test saves hours of dead‑end algebra.

Step‑by‑Step Method to Solve Linear Diophantine Equations

1. Check the gcd condition

• Compute g = gcd(a, b).
• Verify c % g == 0.

Example: 4x + 6y = 7 → gcd(4,6)=2 does not divide 7, so the equation has no integer solutions.

2. Apply the extended Euclidean algorithm

The algorithm returns one pair (x₀, y₀) satisfying a·x₀ + b·y₀ = g.
Multiply both sides by c/g to obtain a particular solution for the original equation.

(a·x₀ + b·y₀) * (c/g) = c

Keep a tiny cheat‑sheet of the back‑substitution steps; it works like magic.

3. Write the parametric form

All solutions are expressed as

x = x₀ + (b/g)·t
y = y₀ - (a/g)·t

where t is any integer. This parametric form lets you generate every possible integer pair.

4. Choose the right integer t

Select t to meet extra constraints (non‑negative values, smallest absolute values, etc.). For a contest you might pick the smallest non‑negative t; for a puzzle you tweak t until the numbers look tidy.

Quick Recap

• GCD test → decides if any solution exists.
• Extended Euclidean algorithm → gives a base solution.
• Parametric form → generates the whole solution set.
• Pick t → satisfies any additional requirements.

Apply these four steps, and you’ll never guess again. Try them on your next homework problem or math competition and watch the “stuck” feeling disappear.