To find three conservative integers \( x, y, z \) such that the equation
\[ 3x + 4y + 5z = 74 \]
holds, we can start by guessing values for \( x \), \( y \), and \( z \) that satisfy that equation.
Let's rearrange this equation to solve for one variable in terms of the others. We can express \( z \) in terms of \( x \) and \( y \):
\[ 5z = 74 - 3x - 4y \implies z = \frac{74 - 3x - 4y}{5} \]
For \( z \) to be an integer, \( 74 - 3x - 4y \) must be divisible by 5. We can check several values for \( x \) and \( y \) that yield integer values of \( z \).
Let's try different pairs of integers for \( x \) and \( y \):
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Let \( x = 4 \), \( y = 10 \): \[ 3(4) + 4(10) + 5z = 74 \implies 12 + 40 + 5z = 74 \implies 5z = 22 \implies z = \frac{22}{5} \quad \text{(not an integer)} \]
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Let \( x = 10 \), \( y = 5 \): \[ 3(10) + 4(5) + 5z = 74 \implies 30 + 20 + 5z = 74 \implies 5z = 24 \implies z = \frac{24}{5} \quad \text{(not an integer)} \]
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Let \( x = 8 \), \( y = 5 \): \[ 3(8) + 4(5) + 5z = 74 \implies 24 + 20 + 5z = 74 \implies 5z = 30 \implies z = 6 \quad \text{(integer)} \]
The values \( x = 8 \), \( y = 5 \), and \( z = 6 \) satisfy the equation.
Verifying:
\[ 3(8) + 4(5) + 5(6) = 24 + 20 + 30 = 74 \]
Thus, the conservative numbers \( x, y, z \) are:
\[ \boxed{(8, 5, 6)} \]