Exercise · ZW-LINEAR-DIOPHANTINE-EQUATION-007 · ★★☆ Intermediate
Given the particular solution \((x_0,y_0)=(2,-1)\) of \(4x+6y=2\), what is the general solution with \(t\in\mathbb{Z}\)?
Choose one option.
Show answer
- A. \(x=2+3t,\ y=-1-2t\)
Explanation
Since \(\gcd(4,6)=2\), the steps are \(6/2=3\) in \(x\) and \(-4/2=-2\) in \(y\). Flipping the sign of the step in \(y\) breaks the equation, and a unit step does not preserve it either.
Review the concept: Linear Diophantine equation
A linear Diophantine equation is \(ax+by=c\) with integer coefficients, where we seek integer pairs \(x,y\) as solutions.
\[x=x_0+\frac{b}{d}t,\quad y=y_0-\frac{a}{d}t,\quad t\in\mathbb{Z}\]
More practice problems: linear diophantine equation
- ZW-LINEAR-DIOPHANTINE-EQUATION-001 What is the greatest common divisor of 12 and 18, which determines whether 12x+18y=c has a solution?
- ZW-LINEAR-DIOPHANTINE-EQUATION-002 Which statement correctly describes when ax+by=c has integer solutions?
- ZW-LINEAR-DIOPHANTINE-EQUATION-003 What is the name of the theorem guaranteeing that d=gcd(a,b) can be written as ax+by with integers…
- ZW-LINEAR-DIOPHANTINE-EQUATION-004 What is the greatest common divisor of 15 and 35?
- ZW-LINEAR-DIOPHANTINE-EQUATION-005 Does the equation 4x+6y=5 have integer solutions?
- ZW-LINEAR-DIOPHANTINE-EQUATION-006 If 7(3)+5(-4)=1, what is a particular solution of 7x+5y=1?
- ZW-LINEAR-DIOPHANTINE-EQUATION-008 Let 9x+15y=12. What is the particular solution obtained from 3=9(2)+15(-1)?
- ZW-LINEAR-DIOPHANTINE-EQUATION-009 For which value of c does the equation 10x+4y=c have an integer solution?
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