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Chapter 3: Problem 33

Solve the equation using the multiplication or division properties of equality. $$-6=-z$$

Short Answer

Expert verified
The solution is \( z = 6 \).

Step by step solution

01

- Understand the Problem

We need to solve the equation \( -6 = -z \) for the variable \( z \). This means we need to find the value of \( z \) that makes the equation true.
02

- Isolate the Variable

To solve for \( z \), you need to get \( z \) by itself on one side of the equation. Since \( z \) is currently multiplied by \( -1 \), we can isolate it by dividing both sides of the equation by \( -1 \).
03

- Divide Both Sides by \( -1 \)

Divide both sides of the equation by \( -1 \):\[-6 \div (-1) = -z \div (-1) \]
04

- Simplify

Simplifying both sides: \(-6 \div (-1) = 6\) and \(-z \div (-1) = z\), resulting in \z = 6\.
05

- Verify the Solution

It is always good practice to check the solution. Substitute \( z = 6 \) back into the original equation: \(-6 = -6\). The solution holds true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication Property of Equality
Understanding and using the multiplication property of equality is crucial when solving linear equations. This property states that the same value can be multiplied to both sides of an equation without changing its equality. For example, if you have an equation like \( a = b \), and you multiply both sides by a non-zero number \( c \), then \( a \times c = b \times c \). This property is often used when you need to eliminate fractions or other divisors attached to the variable. Remember, the key here is that the equality remains unchanged as long as both sides are treated equally.
Division Property of Equality
The division property of equality states that you can divide both sides of an equation by the same non-zero number without changing the equality. This is particularly useful for isolating variables. For example, in the equation \( -6 = -z \), we can divide both sides by \( -1 \) to solve for \( z \). When we do this, we get:
  • \(-6 \div (-1) = -z \div (-1)\)

Simplifying both sides, we get \( 6 = z \). This property makes it simple to move coefficients or other numbers to the opposite side of the equation, helping us isolate and solve for the variable.
Isolating Variables
Isolating the variable in an equation is the key step to solving it. The goal is to get the variable by itself on one side of the equation. To do this, you might need to use the multiplication or division properties of equality. In our example, we started with \( -6 = -z \). To isolate \( z \), we divided both sides by \( -1 \). This removed the negative sign from \( z \), giving us \( z = 6 \).

A common approach to isolating variables includes:
  • Moving terms involving the variable to one side
  • Moving constant terms to the other side
  • Using properties like multiplication or division to simplify
Always remember to perform the same operation on both sides to maintain the equation's balance.

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Most popular questions from this chapter

Comparing the Properties of Equality In your own words, explain how to determine when to use the addition property of equality.

Solve the equation. $$-4 a=0$$

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