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Chapter 11: Problem 6

Evaluate. \(\frac{5 y}{z},\) when \(y=-15\) and \(z=-25\)

Short Answer

Expert verified
The result is \(3\).

Step by step solution

01

Understand the problem

We need to evaluate the expression \(\frac{5 y}{z}\) by substituting the given values for \(y\) and \(z\).
02

Substitute the values

Substitute \(y = -15\) and \(z = -25\) into the expression \(\frac{5 y}{z}\).
03

Perform the substitution

After substitution, the expression becomes \(\frac{5 (-15)}{-25}\).
04

Simplify the expression

Now, simplify the expression: \(\frac{5 \times -15}{-25} = \frac{-75}{-25}\).
05

Simplify the fraction

Since \(\frac{-75}{-25}\) simplifies to \(3\), the final result is \(3\).

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Key Concepts

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

Substitution Method
In algebra, we often need to find the value of an expression by replacing variables with given numbers. This process is known as the substitution method.

To use this method, follow these steps:
  • Identify the variables in the expression.
  • Replace each variable with the provided values.
  • Simplify the expression to find the result.

For example, in the expression \(\frac{5y}{z}\), if we are given that \ y = -15 \ and \ z = -25 \, we replace y and z with these values. Our expression becomes \(\frac{5 \times (-15)}{-25}\).
Substitution makes complex problems easier by converting abstract symbols into concrete numbers.
Simplifying Fractions
Simplifying fractions is a crucial part of evaluating algebraic expressions. It involves reducing the fraction to its simplest form. Here鈥檚 how to do it:
  • Multiply the numerator and the denominator separately if necessary.
  • Identify common factors in the numerator and denominator.
  • Divide both the numerator and the denominator by their greatest common factor (GCF).

Taking our example from earlier, we simplified \(\frac{5 \times -15}{-25}\) to \(\frac{-75}{-25}\).
Then, we divided the numerator and denominator by their GCF, which is -25, resulting in \(\frac{-75\div-25}{-25\div-25} = 3\).
Simplifying fractions helps in breaking down expressions into more manageable and understandable parts.
Negative Numbers
Working with negative numbers can be tricky, but it鈥檚 essential to understand them well in algebra. Here are a few key points:
  • Negative times negative equals positive. So \(-a \times -b = ab\).
  • Negative divided by negative equals positive. Hence, \(\frac{-a}{-b} = \frac{a}{b}\).
  • A negative sign in the numerator or denominator changes the sign of the overall fraction.

In our exercise, we had \(\frac{-75}{-25}\). Dividing negative by negative gives us a positive result, so \(\frac{-75}{-25} = 3\).
Understanding these rules ensures that you handle negative numbers correctly, leading to accurate solutions.

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