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Chapter 3: Problem 15

Solve the equation using the multiplication or division properties of equality. $$-16=-8 z$$

Short Answer

Expert verified
z = 2

Step by step solution

01

Understand the Equation

The given equation is o-16 = -8z. To solve for z, isolate the variable on one side of the equation.
02

Choose the Property

To isolate z, use the division property of equality. This property states that you can divide both sides of the equation by the same non-zero number without changing the equation's equality.
03

Divide Both Sides by -8

Divide both sides of the equation by -8 to isolate z:\[ \frac{-16}{-8} = \frac{-8z}{-8} \]
04

Simplify

Simplify both sides of the equation:\[ 2 = z \]So, z = 2.

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Key Concepts

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

Multiplication Property of Equality
The multiplication property of equality is a key idea when solving linear equations. This property states that if you multiply both sides of an equation by the same non-zero number, the equality remains true. This can be useful when you want to eliminate fractions or when you need to get the variable alone on one side of the equation.
For example, if you have an equation like \( x/4 = 2 \), you can apply the multiplication property of equality by multiplying both sides by 4 to isolate the variable \( x \):
  • Multiply both sides: \( 4 * (x/4) = 2 * 4 \)
  • This simplifies to: \( x = 8 \)
It's an important tool because it allows us to keep the equation balanced and makes it easier to solve for the variable.
Division Property of Equality
The division property of equality helps you solve equations by dividing both sides by the same non-zero number. This action keeps the equation balanced and is particularly helpful when you are trying to isolate a variable that is multiplied by a coefficient.
Consider the original exercise equation: \( -16 = -8z \). To solve for \( z \), you use the division property of equality:
  • Divide both sides by -8: \( \frac{-16}{-8} = \frac{-8z}{-8} \)
  • This simplifies to: \( 2 = z \)
Remember, when you divide by a negative number, the negatives cancel out each other. Both the multiplication and division properties of equality are essential tools for keeping equations balanced while solving for variables.
Isolating the Variable
Isolating the variable means manipulating the equation so that the variable you're solving for ends up alone on one side of the equation. This makes it straightforward to see the solution.
Let's revisit the exercise \( -16 = -8z \). Here is how we isolate the variable \( z \):
  • Recognize that \( z \) is multiplied by \( -8 \).
  • Use the division property of equality to divide both sides by \( -8 \).
  • Simplify: \( \frac{-16}{-8} = \frac{-8z}{-8} \)
    • This calculation gives us \( 2 = z \). Now, \( z \) is isolated, and we can clearly see that \( z = 2 \). This process of isolating the variable is a fundamental step in solving all linear equations.

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