Let's first solve for \( z \) and \( w \) from the equations given.
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From the equation \( 7z = 7 \), divide both sides by 7: \[ z = 1 \]
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From the equation \( 4w = 4 \), divide both sides by 4: \[ w = 1 \]
Now substituting the values of \( z \) and \( w \) into the expression \( 4z + w \): \[ 4z + w = 4(1) + 1 = 4 + 1 = 5 \]
Therefore, the value of the expression \( 4z + w \) is \( \boxed{5} \).