Problem 99 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$\frac{x}{5}-2 y \text { when } x=12 \text { and } y=\frac{4}{5}$$

Step-by-Step Solution

Answer

The evaluated expression equals \(0.8\)

Step 1: Substitute the given values

Substitute \(x = 12\) and \(y = \frac{4}{5}\) into \(\frac{x}{5}-2 y\), which results to \(\frac{12}{5}-2 (\frac{4}{5})\)

Step 2: Perform the Division

First perform the division operation inside the brackets to get \(\frac{12}{5}-2(0.8)\). The result is \(2.4-2(0.8)\).

Step 3: Perform the Multiplication

Perform the multiplication operation to get \(2.4-1.6\).

Step 4: Perform the Subtraction

Finally perform the subtraction operation to get \(0.8\).

Key Concepts

Substitution

Substitution is a fundamental technique in algebra where we replace variables with given numerical values. It's like a simple swapping process. To effectively substitute:

• Identify each variable in the algebraic expression.
• Replace these variables with the specified numerical values.
• Make sure to maintain the structure of the expression.

In the exercise, we substitute the values of \(x = 12\) and \(y = \frac{4}{5}\) into the expression \(\frac{x}{5}-2y\). This substitution transforms the expression into \(\frac{12}{5} - 2\left( \frac{4}{5} \right)\). Understanding substitution ensures that we accurately evaluate any expression by correctly inserting the known values.

Order of Operations

Order of operations is a set of rules that dictate the correct sequence to evaluate a mathematical expression. It's crucial to follow these rules to achieve accurate results. The common acronym used to remember this order is PEMDAS:

• P: Parentheses first.
• E: Exponents (i.e., powers and square roots, etc.)
• M: Multiplication and D: Division (left-to-right)
• A: Addition and S: Subtraction (left-to-right)

In our exercise, after substituting, we follow this order:

• First, perform the division \(\frac{12}{5}\).
• Then, the multiplication of \(2\) and \(\frac{4}{5}\).
• Lastly, subtract the results of these operations: \(2.4 - 1.6\).

Adhering to the correct order of operations is key to solving expressions accurately.

Fractions

Fractions involve numbers expressed as one integer divided by another. In algebra, handling fractions requires care:

• Ensure any substitution results in a fraction being correctly integrated.
• Convert fractions to decimals to simplify operations, if needed.
• Always apply basic operations (addition, subtraction, multiplication, and division) correctly.

In the exercise, we started with fractions \(\frac{12}{5}\) and \(\frac{4}{5}\).
By converting \(\frac{4}{5}\) to a decimal \(0.8\), it simplifies the multiplication with \(2\).
Understanding how to handle fractions, whether through decimal conversion or keeping them as fractions, is crucial to solving algebraic expressions effectively.