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Chapter 15: Problem 12

Solve. $$ \frac{2}{3}=-\frac{z}{8} $$

Short Answer

Expert verified
The solution is \(z = -\frac{16}{3}\).

Step by step solution

01

- Understand the Equation

The given equation is \(\frac{2}{3} = -\frac{z}{8}\). Our goal is to solve for \(z\).
02

- Cross Multiply

To isolate \(z\), cross multiply the fractions. This means multiplying the numerator of the first fraction by the denominator of the second fraction and the denominator of the first fraction by the numerator of the second fraction: \[2 * 8 = -3 * z\]
03

- Simplify the Equation

Simplify the multiplication results: \[16 = -3z\]
04

- Solve for z

To isolate \(z\), divide both sides of the equation by \(-3\): \[z = \frac{16}{-3} = -\frac{16}{3}\]

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Key Concepts

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

cross multiplication
Cross multiplication is a technique used to solve equations that involve fractions. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa.
In our example, we started with the equation \(\frac{2}{3} = -\frac{z}{8}\). Cross multiplying gives us: \[2 \times 8 = -3 \times z \]
This step helps to eliminate the fractions, making it easier to solve for the variable.
Here are tips to ensure accuracy:
  • Always multiply diagonally.
  • Check that you have cross-multiplied both sides evenly.
isolate variables
Isolating variables means rearranging the equation to get the unknown variable on one side of the equation.
After cross multiplying, our equation becomes: \[16 = -3z \]
To isolate \( z \), we need to divide both sides by \( -3 \):
This gives us: \[ z = \frac{16}{-3} \]
Steps to isolate variables:
  • Perform the same operation on both sides of the equation.
  • Use inverse operations to move numbers or coefficients. In this case, we divided by \( -3 \).
simplify fractions
Simplifying fractions involves reducing the fraction to its simplest form. This can make working with the fraction easier.
For our solution, \( z = \frac{16}{-3} = -\frac{16}{3} \), it is already in its simplest form.
To simplify fractions:
  • Find the greatest common divisor (GCD) of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
In our case, 16 and -3 have no common divisors other than 1, so the fraction stays as \( -\frac{16}{3} \).
Remember:
  • A fraction is simplified when no further division can reduce the numerator and denominator.
  • Always simplify if possible, it can make calculations much easier.

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Most popular questions from this chapter

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