- Simplify the left side \(\frac{9z}{9}\) by canceling the common factor of \(9\) in the numerator and denominator.
- Recognize that \(\frac{9z}{9} = 1 \cdot z\).
- Match this to the required form \(\square \cdot z\), where \(\square = 1\).
- Final answer: \(\boxed{1}\).
Explanation
- Simplifying the expression
To simplify the left side of the given equation \(\frac{9z}{9} = \frac{-81}{9}\), we examine the term \(\frac{9z}{9}\). This expression involves a variable \(z\) multiplied by \(9\) in the numerator and divided by \(9\) in the denominator. By the property of division, we know that \(\frac{a \cdot b}{a} = b\) provided that \(a
eq 0\). Here, \(a = 9\) and \(b = z\), so \(\frac{9 \cdot z}{9} = 1 \cdot z\). This allows us to simplify the left side of the equation to \(1 \cdot z\). Comparing this to the desired form \(\square \cdot z\), we identify that the missing value \(\square\) is \(1\).
Examples
In many real-world scenarios, simplifying equations is crucial for solving problems efficiently. For instance, if you are calculating the total cost of multiple items where the unit price is the same, let's say you buy 9 identical notebooks for a total of 81 dollars, you might set up the equation \(\frac{9z}{9} = \frac{81}{9}\), where \(z\) represents the cost of one notebook. By simplifying \(\frac{9z}{9}\) to \(1 \cdot z\), you immediately find that the cost per notebook is \(z = 9\) dollars. This simplification process is the foundation for solving for an unknown in any linear relationship.