Answer:
A) (-17+5k,17-4k)
B) (-4+3k,4-2k)
C) No integer pairs.
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.