We know that x leaves a remainder of 19 when divided by 34, y leaves a remainder of 21, and z leaves a remainder of 29.
Let's start by simplifying the expression (7x - 3y + 5z) by substituting in the remainders we have:
(7x - 3y + 5z) = (7(34a + 19) - 3(34b + 21) + 5(34c + 29))
where a, b, and c are integers representing the number of times each respective number has been divided by 34.
Simplifying this expression, we get:
(7x - 3y + 5z) = (238a + 115 - 102b - 63 + 170c + 145)
Combining like terms, we get:
(7x - 3y + 5z) = (238a - 102b + 170c + 197)
Now we can take this expression and divide it by 17 to find the remainder:
(7x - 3y + 5z) ÷ 17 = (238a - 102b + 170c + 197) ÷ 17
The remainder on the left-hand side will be the same as the remainder on the right-hand side.
We can simplify the right-hand side by dividing each term by 17:
(238a - 102b + 170c + 197) ÷ 17 = (14a - 6b + 10c + 11)
Therefore, the remainder when (7x - 3y + 5z) is divided by 17 is 11.