Use the EXAMPLE 1 as reference by answering the problem, Find all solutions of the Diophantine equation 1245x-365y = 4567 ■ Example 1. Find all possible integral solutions of 18x + 7y=302.Solution: Since (18, 7) = 1 and 1 | 302, the given equation has solutionsNow, we solve for xo, yo such that 18 xo + 7 yo = 118=7 (2) + 47=4 (1) +34= 3(1) + 13= 3(1)Thus we have,1-4-31=18-7(2) - [7 - {18-7(2)}]1 = 18 (2)-7 (5)This means,Xo = 2 andYo=5▪ We substitute these values in the formula. Next. We find all values of t satisfying x>0 and y> 0.604 + 7t > 0and-1510-18t> 0t<-83.9t>-86.3thus, t= -86, -85, -84The table below presents the values of x and y for these values of t usingx = 604 + 7tandy = -1510 - 18ttXу-86-85-84291638202

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Answered: Find all solutions of the Diophantine…

Math

Advanced Math

Find all solutions of the Diophantine equation 1245x-365y = 4567

PREALGEBRA

15th Edition

ISBN:9781938168994

Author:OpenStax

Publisher:OpenStax

Chapter11: Graphs

Section11.1: Use The Rectangular Coordinate System

Problem 11.28TI: Find three solutions to the equation: x + 5y = 10.

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Question

Use the EXAMPLE 1 as reference by answering the problem,

Find all solutions of the Diophantine equation 1245x-365y = 4567

■ Example 1. Find all possible integral solutions of 18x + 7y=302.
Solution: Since (18, 7) = 1 and 1 | 302, the given equation has solutions
Now, we solve for xo, yo such that 18 xo + 7 yo = 1
18=7 (2) + 4
7=4 (1) +3
4= 3(1) + 1
3= 3(1)
Thus we have,
1-4-3
1=18-7(2) - [7 - {18-7(2)}]
1 = 18 (2)-7 (5)
This means,
Xo = 2 and
Yo=5
▪ We substitute these values in the formula. Next. We find all values of t satisfying x>0 and y> 0.
604 + 7t > 0
and
-1510-18t> 0
t<-83.9
t>-86.3
thus, t= -86, -85, -84
The table below presents the values of x and y for these values of t using
x = 604 + 7t
and
y = -1510 - 18t
t
X
у
-86
-85
-84
2
9
16
38
20
2

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