1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
marin [14]
3 years ago
5

For integers a, b, and c, consider the linear Diophantine equation ax C by D c: Suppose integers x0 and y0 satisfy the equation;

that is, ax0 C by0 D c. What other values x D x0 C h and y D y0 C k also satisfy ax Cby D c?
Mathematics
1 answer:
Dmitrij [34]3 years ago
3 0

Answer:

a.

x = x_1+r(\frac{b}{gcd(a, b)} )\\y=y_1-r(\frac{a}{gcd(a, b)} )

b. x = -8 and y = 4

Step-by-step explanation:

This question is incomplete. I will type the complete question below before giving my solution.

For integers a, b, c, consider the linear Diophantine equation

ax+by=c

Suppose integers x0 and yo satisfy the equation; that is,

ax_0+by_0 = c

what other values

x = x_0+h and y=y_0+k

also satisfy ax + by = c? Formulate a conjecture that answers this question.

Devise some numerical examples to ground your exploration. For example, 6(-3) + 15*2 = 12.

Can you find other integers x and y such that 6x + 15y = 12?

How many other pairs of integers x and y can you find ?

Can you find infinitely many other solutions?

From the Extended Euclidean Algorithm, given any integers a and b, integers s and t can be found such that

as+bt=gcd(a,b)

the numbers s and t are not unique, but you only need one pair. Once s and t are found, since we are assuming that gcd(a,b) divides c, there exists an integer k such that gcd(a,b)k = c.

Multiplying as + bt = gcd(a,b) through by k you get

a(sk) + b(tk) = gcd(a,b)k = c

So this gives one solution, with x = sk and y = tk.

Now assuming that ax1 + by1 = c is a solution, and ax + by = c is some other solution. Taking the difference between the two, we get

a(x_1-x) + b(y_1-y)=0

Therefore,

a(x_1-x) = b(y-y_1)

This means that a divides b(y−y1), and therefore a/gcd(a,b) divides y−y1. Hence,

y = y_1+r(\frac{a}{gcd(a, b)})  for some integer r. Substituting into the equation

a(x_1-x)=rb(\frac{a}{gcd(a, b)} )\\gcd(a, b)*a(x_1-x)=rba

or

x = x_1-r(\frac{b}{gcd(a, b)} )

Thus if ax1 + by1 = c is any solution, then all solutions are of the form

x = x_1+r(\frac{b}{gcd(a, b)} )\\y=y_1-r(\frac{a}{gcd(a, b)} )

In order to find all integer solutions to 6x + 15y = 12

we first use the Euclidean algorithm to find gcd(15,6); the parenthetical equation is how we will use this equality after we complete the computation.

15 = 6*2+3\\6=3*2+0

Therefore gcd(6,15) = 3. Since 3|12, the equation has integral solutions.

We then find a way of representing 3 as a linear combination of 6 and 15, using the Euclidean algorithm computation and the equalities, we have,

3 = 15-6*2

Because 4 multiplies 3 to give 12, we multiply by 4

12 = 15*4-6*8

So one solution is

x=-8 & y = 4

All other solutions will have the form

x=-8+\frac{15r}{3} = -8+5r\\y=4-\frac{6r}{3} =4-2r

where r ∈ Ζ

Hence by putting r values, we get many (x, y)

You might be interested in
4. Convert 23 miles per hour to kilometers per minute. (1 mile = 1.609 kilometers)
Karo-lina-s [1.5K]

Answer:

Ewan ko lng po when you find someone ma mas better less sa kanya

8 0
2 years ago
PLEASEEE help with 3-4 and a-b :)
attashe74 [19]
The awnser would be-(4a+b-3) I hope this helps you
3 0
2 years ago
A model car is 8 inches long and 5.2
Sphinxa [80]
Use proportions here - 8/5.2 = 96/width of actual car. using cross multiplication, 5.2*96 = 8x. this gives you 499.2 = 8x. divide both sides by eight to get x=62.4. the width of the actual car is 62.4 inches.
7 0
2 years ago
Please help thanks! Brainliest <br> Sorry for spamming.
Bad White [126]

Answer: −10

Step-by-step explanation: 9^2−21/−6

81−21/−6

60/−6

−10

6 0
3 years ago
The equation is solved. Describe the property used in each solution.
UNO [17]

Answer:

Step-by-step explanation:

It’s multiplication and then division. Just did it on edge2020.

7 0
3 years ago
Other questions:
  • A Lights-A-Lot quality inspector examines a sample of 30 strings of lights and finds that 8 are defective.
    5·1 answer
  • I will give brainliest
    9·2 answers
  • A circular kitchen table has a diameter of 60 inches how much fabric is needed to cover the entire table
    6·2 answers
  • What is 0.624 an expanded form
    15·1 answer
  • [super easy] i have to draw an ugly sweater based around the vocab term natural selection, any ideas?
    12·1 answer
  • Help please I need this for my homework
    14·1 answer
  • (1/2+2/3)÷2 5/6×15-3 1/7​
    7·1 answer
  • Hi math gives me big sad
    10·2 answers
  • N less than 10 <br> .......................
    5·1 answer
  • Four students are trying to find the rule that translates point N(–2, –4) to N prime (2, 4). Each student’s reasoning is shown b
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!