Problem 16 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when x = 3. $$ \frac{24}{x} \cdot 5 $$

Step-by-Step Solution

Answer

The value of the expression \(\frac{24}{x} \cdot 5\) when \(x = 3\) is 40.

Step 1: Substitution

Substitute 3 for \(x\) in the expression \(\frac{24}{x} \cdot 5\). That gives you \(\frac{24}{3} \cdot 5\).

Step 2: Evaluate Division

Evaluate the division operation first. \( \frac{24}{3} \) equals 8. So, the expression becomes \(8 \cdot 5\).

Step 3: Evaluate Multiplication

Now, evaluate the multiplication: \(8 \cdot 5\) equals 40.

Key Concepts

Substitution Method

The substitution method is an essential tool when evaluating variable expressions. It involves replacing a variable in an expression with a given value. Whenever you are provided with a specific value for a variable, your first step should be to substitute this value into your expression.
This allows you to transform an algebraic expression into a numerical one that is easier to evaluate.
In the context of our exercise, we replace "x" with 3 in the expression \( \frac{24}{x} \cdot 5 \). The expression then becomes \( \frac{24}{3} \cdot 5 \).
Keep in mind the importance of carefully substituting each variable. Even a small error will lead to incorrect evaluations. After substitution, what follows next is the application of the correct mathematical operations to simplify your expression further.

Order of Operations

The order of operations is a crucial principle in mathematics that dictates the sequence in which different operations should be carried out to ensure accurate results. Without this standardized order, the same expression could yield different outcomes when computed by different people. To remember this order, you can use the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In our exercise, we apply this order to first perform the division \( \frac{24}{3} \), since division comes before multiplication.
This results in 8. After resolving all the operations of higher precedence, you then proceed to multiplication.
Adhering to this order is vital to obtaining the correct answer, and overlooking it might lead to errors.

Division and Multiplication in Expressions

Understanding how to handle division and multiplication within expressions is key to solving them accurately. Once you have substituted your variables and organized your operations according to their correct order, you'll often find division and multiplication as the next steps.
If division and multiplication appear together as in \( \frac{24}{3} \cdot 5 \), you should handle them respectively as they appear from left to right.
In our example, we first solve \( \frac{24}{3} = 8 \), and then multiply the result by 5, leading to the final solution of 40. It’s important to complete each operation in sequence to avoid mistakes.
By clearly understanding and practicing these processes, you'll become more adept at evaluating expressions confidently.