Problem 15 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when x = 3. $$ \frac{15}{x}+2^{3}-10 $$

Step-by-Step Solution

Answer

The evaluated expression gives us the result 3.

Step 1: Substitute the Given Value

In this step, replace the variable x with the given value 3 in the expression. So, the expression \( \frac{15}{x}+2^{3}-10 \) becomes \( \frac{15}{3}+2^{3}-10 \).

Step 2: Application of Operations

Apply the operations in the correct order, that is, firstly evaluate the power, then division, and lastly subtraction. The expression becomes \( 5+8-10 \).

Step 3: Evaluate the Expression

Now, solve the simple arithmetic to find the value of the expression. When we subtract 10 from the result of addition, the expression becomes \( 13-10 = 3\).

Key Concepts

Substitute Variable Values

When evaluating variable expressions, a critical first step is substituting variable values. This involves replacing the variable in an expression with its corresponding numerical value.

For instance, given the expression \( \frac{15}{x}+2^{3}-10 \) and the information that \( x = 3 \) you'll swap every \( x \) with 3, resulting in \( \frac{15}{3}+2^{3}-10 \).

Why Substitute?

Substitution is essential because it transforms an abstract algebraic expression into a concrete numerical one that can be solved using basic arithmetic. Without this critical step, it would be impossible to evaluate the expression for a specific value of the variable.

Order of Operations

The order of operations is the sequence in which an expression with more than one operation should be evaluated. It's crucial for avoiding errors and reaching the correct result.

The standard order is abbreviated as PEMDAS:

• Parentheses
• Exponents (i.e., powers and roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this sequence ensures accurate calculations. For the expression \( \frac{15}{3}+2^{3}-10 \), you must first calculate the exponent \( 2^{3} \), then perform the division \( \frac{15}{3} \), and finally, carry out the subtraction (10).

Remembering the Order

You can remember the order with the phrase 'Please Excuse My Dear Aunt Sally' or any other mnemonic that helps recall the sequence.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division—the fundamental building blocks of math. Once variable values are substituted and the correct order of operations is determined, you perform these basic calculations to solve the expression.

In our example, after substituting and applying the order of operations, we perform the division \( 15 \div 3 = 5 \) followed by the exponentiation \( 2^{3} = 8 \) and finally the subtraction \( 5 + 8 - 10 \). This sequential application of arithmetic operations simplifies the expression to \( 13 - 10 \), yielding a final result of 3.

Arithmetic Checks

Double-checking each arithmetic step can help avoid simple errors that may lead to incorrect answers, ensuring the integrity of the final result.