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Chapter 5: Problem 88

Evaluate each expression for \(x=6, y=0.3,\) and \(z=-2.4 .\) $$ \frac{x}{y}+2 z $$

Short Answer

Expert verified
The value of the expression is 15.2.

Step by step solution

01

Substitute the Values into the Expression

Substitute the given values of the variables into the expression \( \frac{x}{y} + 2z \). Thus, replace \(x\) with 6, \(y\) with 0.3, and \(z\) with -2.4. This gives us the expression: \( \frac{6}{0.3} + 2(-2.4) \).
02

Calculate the Division

Perform the division \( \frac{6}{0.3} \). To do this, divide 6 by 0.3, which is equivalent to multiplying 6 by \( \frac{10}{3} \). \( 6 \times \frac{10}{3} = 20 \). So, \( \frac{6}{0.3} = 20 \).
03

Calculate the Multiplication

Compute the second part of the expression, \( 2z \), by substituting the value of \(z\). We have \( 2(-2.4) = -4.8 \).
04

Add the Results

Now, add the results of Step 2 and Step 3. Thus, the expression becomes \( 20 - 4.8 \). Calculating this, we get \( 15.2 \).

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Key Concepts

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

Variable Substitution
Variable substitution is the first step in evaluating expressions where specific values are given for each variable. Imagine each variable as a placeholder that needs to be replaced.
For example, consider the expression \( \frac{x}{y} + 2z \). When we're given particular values for \( x \), \( y \), and \( z \), we replace or substitute these letters with the given numbers.
This allows us to simplify and evaluate the expression effectively. In our case:
  • Replace \( x \) with 6
  • Replace \( y \) with 0.3
  • Replace \( z \) with -2.4
The expression morphs into: \( \frac{6}{0.3} + 2(-2.4) \).By substituting these values, we set the stage for further operations.
Basic Arithmetic Operations
Basic arithmetic operations are the core methods we use to transform and solve numerical expressions. These operations include addition, subtraction, multiplication, and division.
In our expression \( \frac{6}{0.3} + 2(-2.4) \), we perform two main operations:
  • **Division**: Dividing 6 by 0.3, understood as multiplying by the reciprocal. This is because dividing by a number is the same as multiplying by its inverse.
    Hence, \( 6 \div 0.3 = 6 \times \frac{10}{3} = 20 \).

  • **Multiplication**: Multiplying 2 by -2.4. This simply means doubling -2.4 or adding -2.4 to itself.
    This gives us \( 2 \times -2.4 = -4.8 \).
Combining these operations is essential for moving forward to the final computation.
Order of Operations
The order of operations is vital when tackling mathematical expressions, ensuring each part of the problem is solved in the right sequence.
You might remember the acronym PEMDAS:
  • **P**arentheses
  • **E**xponents
  • **M**ultiplication and **D**ivision (from left to right)
  • **A**ddition and **S**ubtraction (from left to right)
For the expression \( \frac{6}{0.3} + 2(-2.4) \), we follow the order:
  • First, handle the division: \( \frac{6}{0.3} \) is calculated as 20.
  • Next, perform the multiplication: \( 2 \times -2.4 \) results in -4.8.
  • Finally, add the results together: \( 20 + (-4.8) = 15.2 \).
By following these steps, order is maintained, ensuring the expression is evaluated correctly.

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