Year 11 Maths - Specialist Number systems and sets

Linear Diophantine Equations

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Questions

Question 1

30029

Find all the solutions to the Diophantine equation \(5x + 3y = 1\)

Answer

\(x = 2 + 3t,{\rm{ }}y = - 3 - 5t\)

Worked Solution

\begin{align}
&5 x+3y=1 {\color{red}\ldots(1)}\\
&x=x_{0}+3t, \quad y=y_{0}-5t\\
&\text{Let}\ x=2\ \text{in}\ {\color{red}(1)}\\
&\begin{aligned} 10+3 y &=1 \\ 3 y &=-9 \\ y &=-3 \end{aligned}
\end{align}

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