To find the value of \( z \) that makes the expressions \( 4x - 5y \) and \( (3y + 2x) - z(4y - x) \) equivalent, we can set them equal to each other and solve for \( z \).
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Start by expressing the condition: \[ 4x - 5y = (3y + 2x) - z(4y - x) \]
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Distribute \( z \) in the expression on the right: \[ 4x - 5y = 3y + 2x - z(4y) + z(x) \] \[ 4x - 5y = 3y + 2x + zx - 4zy \]
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Rearrange to bring everything to one side: \[ 4x - 5y - 3y - 2x + 4zy - zx = 0 \] \[ (4x - 2x - zx) + (-5y - 3y + 4zy) = 0 \]
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This simplifies to: \[ (2 - z)x + (-8 + 4z)y = 0 \]
For this equation to hold for all \( x \) and \( y \), both coefficients must equal zero:
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First coefficient: \[ 2 - z = 0 \implies z = 2 \]
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Second coefficient: \[ -8 + 4z = 0 \implies 4z = 8 \implies z = 2 \]
Both conditions yield \( z = 2 \).
Thus, the value of \( z \) that makes the expressions equivalent is: \[ \boxed{2} \]