Problem 58 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$\frac{16}{x}-3 x$$

Step-by-Step Solution

Answer

The expression evaluates to 8 when \(x = -4\).

Step 1: Substitute the Value of x

Begin by substituting the given value of \(x = -4\) into the expression \(\frac{16}{x} - 3x\). This gives us: \(\frac{16}{-4} - 3(-4)\).

Step 2: Perform Division

Calculate \(\frac{16}{-4}\). Since \(16 \div -4 = -4\), replace \(\frac{16}{-4}\) with \(-4\).

Step 3: Multiply the Terms

Now compute \(-3(-4)\). The product of two negative numbers is positive, so \(-3 \times -4 = 12\).

Step 4: Simplify the Expression

Combine the results from the previous steps to simplify the expression: \(-4 + 12\).

Step 5: Calculate the Final Result

Simplify \(-4 + 12\) to find the value of the expression. \(-4 + 12 = 8\).

Key Concepts

Substitution Method

Substitution is a common technique used in algebra to solve expressions and equations. It involves replacing a variable in an expression with a specific value. This helps simplify the problem and makes it easier to solve.
For our exercise, we are given the expression \( \frac{16}{x} - 3x \) with \( x = -4 \). When using substitution, you start by replacing every instance of the variable \( x \) in the expression with \( -4 \). So the expression becomes:

• \( \frac{16}{-4} - 3(-4) \)

This conversion lets us proceed with numeric calculations rather than working with abstract variables. All of the subsequent operations become standard arithmetic steps.
This method is vital not just for evaluating expressions but also for solving equations when an exact value of a variable is known.

Arithmetic Operations

Arithmetic operations are essential in solving mathematical problems. These basic operations include addition, subtraction, multiplication, and division. They are used to manipulate numbers and are the foundation of mathematics.
In our example, there are two main types of operations to perform :

• Division: First, we need to solve the division part of \( \frac{16}{-4} \). Dividing a positive number by a negative one gives a negative result, so \( 16 \div -4 = -4 \).
  
• Multiplication: Next, we compute \(-3 \times -4 \). Multiplying two negative numbers yields a positive result, so \(-3 \times -4 = 12 \).

After tackling each operation separately, the results can be combined in the final step to find the total value of the expression. Each operation needs to be carefully executed to ensure accuracy.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a particular value or set of values. These can be as straightforward as a single number or variable, or as complex as formulas involving various operations.
In the case of our expression \( \frac{16}{x} - 3x \), there are several elements to consider:

• Numbers: \(16\) is a constant number, serving as a part of the division operation.
• Variable: \(x\) is the variable that we substitute with a specific number. In this specific context, \(x = -4\).
• Operations: Division (\( \frac{16}{x} \)) and multiplication (\(-3x\)) followed by subtraction combine to form the complete expression.

Understanding how these components interact and how to manipulate them is crucial for solving mathematical expressions. By learning to break down and substitute within these expressions, students develop the skills necessary for more advanced mathematical problem-solving.