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Chapter 5: Problem 321

In the following exercises, solve the equation. $$ \frac{p}{-5}=-1.65 $$

Short Answer

Expert verified
p = 8.25

Step by step solution

01

Understand the equation

The given equation is a simple linear equation: \[ \frac{p}{-5} = -1.65 \]. The goal is to solve for the variable \( p \).
02

Isolate the variable

To isolate \( p \), multiply both sides of the equation by \( -5 \). This will cancel out the denominator on the left side of the equation. \[ \frac{p}{-5} \times (-5) = -1.65 \times (-5) \] Simplifying this gives: \[ p = 8.25 \]
03

Check the solution

To ensure the solution is correct, substitute \( p = 8.25 \) back into the original equation: \[ \frac{8.25}{-5} \] should equal \( -1.65 \). Performing the division gives: \[ -1.65 = -1.65 \]. Since the left-hand side equals the right-hand side, the solution is correct.

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Key Concepts

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

isolating variables
Isolating variables is crucial when solving equations because it helps you find the value of the unknown. In our equation, \(\frac{p}{-5} = -1.65\), we need to isolate \( p \) to solve for it. To do this, we eliminate the fraction by multiplying both sides by -5. This step is essential because it removes the denominator and allows us to work with the variable more easily. When isolating variables, always perform the same operation on both sides of the equation to maintain balance. This ensures that the equation remains equal.
multiplication in equations
Multiplication is a powerful tool in equations, especially when dealing with fractions. In our given equation, \(\frac{p}{-5} = -1.65\), multiplying by -5 on both sides helps isolate \( p \). Let's break it down:
\(\frac{p}{-5} \times (-5) \) cancels out the -5 in the denominator, leaving us with \( p \). On the right-hand side, we have \(-1.65 \times (-5) \), which simplifies to 8.25. This operation is straightforward and necessary for simplifying the equation. Remember:
  • Always multiply both sides equally.
  • Check your multiplication steps to avoid mistakes.
By following these guidelines, multiplying in equations becomes a reliable method for isolating variables.
verification of solutions
Verification ensures that the solution you've found is accurate. For our exercise, after isolating \( p \) and finding \( p = 8.25\), we substitute it back into the original equation to check our work. Let's verify:
Substituting, \(\frac{8.25}{-5} \) should equal \(-1.65\). Performing the division results in:
\(-1.65 = -1.65\)
Since both sides of the equation match, our solution is verified. This step is important because it confirms the correctness of our work. Always verify:
  • Re-substitute the solution.
  • Ensure the left and right sides of the equation are equal.
Verification gives you confidence in your solution and ensures accuracy.

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