To determine the value that correctly fills the blank so that the expressions are equivalent, we first evaluate the left and right sides of the equation as follows:
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Calculate the left side: \[ 42 + 70 = 112 \]
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Calculate the right side expression: Assuming the blank represents an unknown value \(x\), we compute: \[ x \cdot (6 + 10) = x \cdot 16 \]
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Set both sides equal: \[ 112 = x \cdot 16 \]
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Solve for \(x\): To solve for \(x\), divide both sides by 16: \[ x = \frac{112}{16} = 7 \]
Now that we have solved for \(x\), we can evaluate the choices:
- 10: Not equal to \(7\).
- 7: Correct because we found \(x = 7\).
- 6: Not equal to \(7\).
- 20: Not equal to \(7\).
Thus, the correct value that fills in the blank is: \[ \boxed{7} \]