Solve the given linear Diophantine equation. Show all necessary work. A) 4x + 5y=17 B) 6x+9y=12 C) 4x+10y=9

A) (-17+5k, 17-4k)

B) (-4+3k, 4-2k)

C) No integer pairs.

Step-by-step explanation:

To do this, I'm going to use Euclidean's Algorithm.

4x+5y=17

5=4 (1) + 1

4=1 (4)

So going backwards through those equations:

5-4 (1) = 1

-4 (1) + 5 (1) = 1

Multiply both sides by 17:

4 (-17) + 5 (17) = 17

So one integer pair satisfying 4x+5y=17 is (-17,17).

What is the slope for this equation?

Let's put it in slope-intercept form:

4x+5y=17

Subtract 4x on both sides:

5y=-4x+17

Divide both sides by 5:

y = (-4/5) x + (17/5)

The slope is down 4 and right 5.

So let's show more solutions other than (-17,17) by using the slope.

All integer pairs satisfying this equation is (-17+5k, 17-4k).

Let's check:

4 (-17+5k) + 5 (17-4k)

-68+20k+85-20k

-68+85

17

That was exactly what we wanted since we were looking for integer pairs that satisfy 4x+5y=17.

Onward to the next problem.

6x+9y=12

9=6 (1) + 3

6=3 (2)

Now backwards through the equations:

9-6 (1) = 3

9 (1) - 6 (1) = 3

Multiply both sides by 4:

9 (4) - 6 (4) = 12

-6 (4) + 9 (4) = 12

6 (-4) + 9 (4) = 12

So one integer pair satisfying 6x+9y=12 is (-4,4).

Let's find the slope of 6x+9y=12.

6x+9y=12

Subtract 6x on both sides:

9y=-6x+12

Divide both sides by 9:

y = (-6/9) x + (12/9)

Reduce:

y = (-2/3) x + (4/3)

The slope is down 2 right 3.

So all the integer pairs are (-4+3k, 4-2k).

Let's check:

6 (-4+3k) + 9 (4-2k)

-24+18k+36-18k

-24+36

12

That checks out since we wanted integer pairs that made 6x+9y=12.

Onward to the last problem.

4x+10y=9

10=4 (2) + 2

4=2 (2)

So the gcd (4,10) = 2 which means this one doesn't have any solutions because there is no integer k such that 2k=9.