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$9 z=-81$



The Multiplication Property of Equality states that if $a, b$, and $c$ represent numbers and if $c
eq 0$ and $a = b$, then the following are true:

$a \cdot c=b \cdot c \text { and } \frac{a}{c}=\frac{b}{c}$



To get $z$ by itself, use the multiplication property of equality to divide both sides by 9.

$\begin{array}{l}

9 z=-81 \\

\frac{9 z}{\square}=\frac{-81}{\square}

\end{array}$

Verified Answer :

  • Identify the given linear equation \(9z = -81\).
  • Apply the Division Property of Equality by dividing both sides of the equation by 9.
  • Simplify the expression to find the value of the variable.
  • The result is \(z = \boxed{-9}\).

Explanation

  1. Analysis of the equation

The objective is to isolate the variable \(z\) in the given equation \(9z = -81\). To do this, we need to apply the Division Property of Equality, which is a corollary of the Multiplication Property of Equality. This property states that if we divide both sides of an equation by the same non-zero number, the equality remains true. Here, the variable \(z\) is being multiplied by 9 on the left-hand side. To isolate \(z\), we must perform the inverse operation, which is division by 9.

  1. Isolating the variable z

To solve for \(z\), we divide both sides of the equation by 9:
\[\frac{9z}{9} = \frac{-81}{9}\]
By simplifying the left side, we get \(\frac{9}{9} \times z = 1 \times z = z\).
On the right side, we perform the division: \(-81 \div 9 = -9\).
Thus, we arrive at the result:
\[z = -9\]

Examples

Imagine you have a debt of \(\$81\) that needs to be paid off in 9 equal monthly installments. To find the amount of each monthly payment, you would divide the total debt by the number of installments: \(\frac{-81}{9} = -9\). In this context, \(z = -9\) represents the amount of money deducted from your balance each month. Keeping track of such values helps in maintaining a balanced budget.

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