Exercise · ZW-LINEAR-DIOPHANTINE-EQUATION-010 · ★★★ Advanced
What particular solution of \(21x+14y=35\) comes from \(7=21(1)+14(-1)\)?
Choose one option.
Show answer
- B. \((x,y)=(5,-5)\)
Explanation
Since \(\gcd(21,14)=7\) divides \(35\), multiply the identity \(7=21(1)+14(-1)\) by \(5\) to get \((5,-5)\), which satisfies \(21(5)+14(-5)=35\).
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-007 Given the particular solution (x_0,y_0)=(2,-1) of 4x+6y=2, what is the general solution with t∈ℤ?
- ZW-LINEAR-DIOPHANTINE-EQUATION-008 Let 9x+15y=12. What is the particular solution obtained from 3=9(2)+15(-1)?