Problem 57 - Prealgebra

Question

Evaluate each expression if \(a=6, b=4,\) and \(c=5\). $$4 a-(b+c)$$

Step-by-Step Solution

Answer

The value of the expression is 15.

Step 1: Substitute the variables

Firstly, substitute the given values of the variables into the expression. For the expression \(4a - (b+c)\), substitute \(a = 6\), \(b = 4\), and \(c = 5\). The expression becomes \(4(6) - (4 + 5)\).

Step 2: Perform multiplication

Next, perform the multiplication in the expression. Calculate \(4 \times 6 = 24\). The expression now reads \(24 - (4 + 5)\).

Step 3: Add inside the parentheses

Then, add the numbers inside the parentheses. Calculate \(4 + 5 = 9\). The expression now becomes \(24 - 9\).

Step 4: Subtract

Finally, perform the subtraction to evaluate the expression. Subtract \(9\) from \(24\): \(24 - 9 = 15\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a technique used to replace variables with their given numerical values. This is especially useful in simplifying expressions and solving equations. When given an expression, like in the problem \[ 4a - (b + c), \] we are informed that the variables have specific values:

• \( a = 6 \)
• \( b = 4 \)
• \( c = 5 \)

Substitution involves directly replacing each variable in the expression with these numbers. This step transforms the abstract expression into a numerical form that can be easily evaluated using basic arithmetic operations.
For our expression, substitution changes it to:
\[ 4(6) - (4 + 5). \] After substitution, solving the expression becomes simpler, as it now only requires basic arithmetic.

Order of Operations

The order of operations is a fundamental concept in algebra that dictates the correct sequence in which mathematical operations must be performed to accurately simplify an expression.
This is often remembered by the acronym PEMDAS:

• **P**: Parentheses first
• **E**: Exponents (none in this particular problem)
• **MD**: Multiplication and Division (from left to right)
• **AS**: Addition and Subtraction (from left to right)

Applying this rule to the expression \[ 4(6) - (4 + 5), \] we start with the operations inside the parentheses:
Add \(4 + 5\) to get \(9\).
Next, follow with multiplication:
Calculate \(4 \times 6\) which is \(24\).
Lastly, perform the subtraction:
Subtract \(9\) from \(24\), resulting in \(15\).
Each step respects the operations hierarchy, ensuring the solution's correctness.

Basic Arithmetic Operations

At the core of simplifying expressions through substitution and order of operations are basic arithmetic operations: addition, subtraction, multiplication, and division. These operations form the backbone of prealgebra and algebra, allowing us to calculate and simplify expressions.
Using the exercise as an example, let's explore how each operation was applied:

• **Multiplication**: We first executed multiplication within the substitution: \(4 \times 6\) translating to \(24\).
• **Addition**: Addition was performed next inside the parentheses: \(4 + 5\), which adds up to \(9\).
• **Subtraction**: Finally, subtraction simplified the overall expression with \(24 - 9\), leading us to the result of \(15\).

By mastering these basic arithmetic operations, solving more complex algebraic expressions becomes straightforward. Consistently applying these steps ensures accurate results in any mathematical problem.