To complete the expression
\[ \frac{2t}{7t - 28} = \frac{\text{nothing}}{21t - 84}, \]
we first simplify both sides.
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Simplify the left side:
The denominator on the left, \(7t - 28\), can be factored: \[ 7t - 28 = 7(t - 4). \] Thus, we have: \[ \frac{2t}{7(t - 4)}. \]
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Simplify the right side:
The denominator on the right, \(21t - 84\), can also be factored: \[ 21t - 84 = 21(t - 4). \] Therefore, we rewrite the right side as: \[ \frac{\text{nothing}}{21(t - 4)}. \]
Now we need to equate the two sides: \[ \frac{2t}{7(t - 4)} = \frac{\text{nothing}}{21(t - 4)}. \]
To find what "nothing" corresponds to, we can cross-multiply: \[ 2t \cdot 21(t - 4) = \text{nothing} \cdot 7(t - 4). \]
To solve for "nothing," we note that if we examine the relationship revealed by cross-multiplying: \[ \text{nothing} = \frac{2t \cdot 21(t - 4)}{7(t - 4)}. \] Since \(t - 4\) cancels out (as long as \(t \neq 4\)), we have: \[ \text{nothing} = \frac{2t \cdot 21}{7} = 6t. \]
Finally, we can express the completed and simplified original fraction equality as: \[ \frac{2t}{7t - 28} = \frac{6t}{21t - 84}. \]
So, the completed expression would conclude with: \[ \text{nothing} = 6t. \]