Givens
Two odd integers which can be adjusted to make 194
Let the smaller integer = 2x + 1 This guarantees that it is odd.
Let the larger integer = 2x + 3
Solve
Four times the larger = 4(2x + 3)
Twice the smaller = 2(2x + 1)
Sum of both = 4(2x + 3) + 2(2x + 1) = 194 Remove the brackets.
8x + 12 + 4x + 2 = 194 Collect Like terms
12x + 14 = 194 Subtract 14 from both sides
12x = 194 - 14
12x = 180 Divide by 12
x = 180/12
x = 15
Answer
The smaller integer = 2*15 + 1 = 30 + 1 = 31
The larger integer = 2*15 + 3 = 30 + 3 = 33
Check
Four times the larger integer = 4*33 = 132
Two times the smaller integer =2*31 = 62
Sum 194. It checks.
Comment
I'm not sure you have to do this, but it is safest to make sure that you are adding two consecutive odd integers together. That's why we started out with 2x because that is even and then added 1 or 3 to it to make it odd.