Problem 6 - Prealgebra

Question

Evaluate each expression if \(a=5, b=12,\) and \(c=4\) $$5 a-(b-c)$$

Step-by-Step Solution

Answer

The value of the expression is 17.

Step 1: Substitute the Values

Begin by substituting the given values into the expression. Replace \(a\) with 5, \(b\) with 12, and \(c\) with 4 in the expression \(5a - (b - c)\). This gives us \(5(5) - (12 - 4)\).

Step 2: Calculate Multiplication

Calculate the multiplication part of the expression. Multiply 5 by 5 to get 25. Now, the expression is \(25 - (12 - 4)\).

Step 3: Calculate the Parenthesis

Next, solve the expression inside the parenthesis: \(12 - 4\). Subtract 4 from 12, which equals 8. Now the expression is \(25 - 8\).

Step 4: Solve the Subtraction

Finally, subtract 8 from 25 which equals 17.

Key Concepts

Understanding Substitution in Algebra

Substitution is a fundamental concept in algebra. In essence, it means replacing a variable with a specific value. By doing this, we transform an algebraic expression into a simpler arithmetic expression that can be calculated directly.

Let's consider the original exercise: to evaluate the expression \(5a - (b - c)\) with given values \(a=5, b=12,\) and \(c=4\). To apply substitution here, follow these steps:

• Identify each variable in the expression.
• Replace \(a\) with 5, \(b\) with 12, and \(c\) with 4.
• Rewrite the expression: \(5(5) - (12 - 4)\).

This transformation allows you to proceed with straightforward arithmetic calculations, using known numerical values instead of abstract variables.

The Role of Parentheses in Expressions

Parentheses play a critical role in algebraic expressions. They indicate which operations should be performed first and help in organizing expressions, especially when multiple operations are involved.

In the given exercise, the expression \(5a - (b - c)\) contains parentheses around \(b - c\). This determines that the subtraction \(b - c\) needs to be evaluated before any operation outside of the parentheses. Here’s how it works:

• First, solve any operation inside the parentheses. In this case, subtract 4 from 12 to get 8.
• This step transforms the expression to \(5(5) - 8\).

Ignoring parentheses would lead to incorrect results, emphasizing their importance in maintaining the correct order of operations.

Mastering the Order of Operations

The order of operations is vital in ensuring that calculations are done consistently and correctly. It’s typically remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

In our exercise, after substitution, the expression is \(5(5) - (12 - 4)\). Based on the order of operations, we first perform the calculation inside the parentheses, reducing \(12 - 4\) to 8. Next, we deal with multiplication: \(5(5)\) becomes 25.

Following this, we proceed to subtraction, the final operation: subtract 8 from 25 to arrive at the solution, 17. Always remember, adhering to the correct order of operations is crucial for accurate results in any mathematical problem.