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Chapter 4: Problem 670

In the following exercises, translate and solve. The quotient of \(p\) and \(-4\) is \(-8 .\)

Short Answer

Expert verified
The solution is \(p = 32\).

Step by step solution

01

Translate the problem into an equation

The phrase 'The quotient of p and -4 is -8' translates to \ \ \(\frac{p}{-4} = -8\).
02

Multiply both sides by -4

To isolate the variable \(p\), multiply both sides of the equation by -4: \ \ \(\frac{p}{-4} \times -4 = -8 \times -4\).
03

Simplify the equation

Upon simplifying, we get: \ \ \(p = 32\).
04

Verify the solution

To ensure the solution is correct, substitute \(p = 32\) back into the original equation: \ \ \(\frac{32}{-4} = -8\). \ \ Since this is true, our solution \(p = 32\) is correct.

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Key Concepts

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

Understanding Algebra Translation
In algebra, 'translation' refers to converting a word problem into a mathematical equation. For example, let's look at the phrase from our original exercise: 'The quotient of p and -4 is -8.' Here, 'quotient' means division. So, we translate this to the equation: \( \frac{p}{-4} = -8 \). This is a crucial step because having the correct equation is the foundation for solving the problem.
Tips for translating words to equations:
  • Identify keywords like 'sum,' 'difference,' 'product,' and 'quotient.'
  • Replace these words with the corresponding mathematical operations.
  • Make sure all quantities and variables are correctly represented.
Mastering the Art of Isolating Variables
Next, let's talk about isolating variables. To solve for a variable, we need to get it by itself on one side of the equation. Looking at our equation: \( \frac{p}{-4} = -8 \), we aim to isolate p. We do this by performing the inverse operation. Since p is divided by -4, we multiply both sides by -4 to cancel out the division: \( \frac{p}{-4} \times -4 = -8 \times -4 \). This simplifies to \( p = 32 \).
Important steps to isolate variables:
  • Identify the operation keeping the variable from being isolated (addition, subtraction, multiplication, or division).
  • Apply the inverse operation to both sides of the equation.
  • Perform the simplification to isolate the variable.
Verifying Solutions for Accuracy
Verification is the final step to ensure that our solution is correct. This involves substituting the found value back into the original equation to see if it satisfies the equation. Plugging our solution p = 32 back into \( \frac{p}{-4} = -8 \), we get \( \frac{32}{-4} = -8 \). Since \( -8 = -8 \) holds true, our solution is indeed correct.
Key points for verifying solutions:
  • Substitute the solution back into the original equation.
  • Check if both sides of the equation are equal.
  • If they are, you've found the correct solution; otherwise, recheck your steps.

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