Problem 12 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=4\). $$5(x+3)$$

Step-by-Step Solution

Answer

The given expression equals 35 when \(x = 4\).

Step 1: Substitution

Replace the variable \(x\) in the given expression with the value given, which is 4. The expression becomes \(5*(4+3)\).

Step 2: Parentheses

Perform the operation inside the parentheses first. \(4+3 = 7\). The expression now becomes \(5*7\).

Step 3: Multiplication

Perform the multiplication. \(5*7 = 35\). Therefore, this expression equals to 35 when \(x = 4\).

Key Concepts

Evaluating Expressions

Evaluating expressions is a fundamental concept in algebra. It involves calculating the value of an algebraic expression by replacing variables with numbers.
For example, in the expression \(5(x + 3)\), "\(x\)" is a variable that stands in for a number. By evaluating the expression, you determine what the numeric value is when specific numbers replace the variables.
This process helps simplify and solve equations, which is integral to solving math problems in both an academic setting and real-life situations.

• The first step in evaluating an expression is understanding the variables involved.
• Next, identify what number or value each variable will be replaced with.
• Lastly, follow the mathematical operations required to reach a solution.

Substitution Method

The substitution method is a technique used to evaluate expressions by replacing variables with given numbers.
This method makes solving an expression more manageable and less abstract.
For the expression \(5(x + 3)\) when \(x = 4\), the substitution step involves replacing \(x\) with 4, transforming the expression to \(5(4 + 3)\).
After substituting, the expression becomes much simpler to deal with as you then proceed with arithmetic operations.

• Begin by identifying the variable in the expression.
• Substitute the variable with its given numeric value.
• Proceed to simplify the expression step-by-step.

Substitution is especially powerful as it can be utilized with many variables and more complex expressions, not just single operations.

Order of Operations

The order of operations is essential when evaluating expressions. It dictates the sequence in which operations should be performed to ensure consistent and correct results.
A common acronym to remember the order is PEMDAS:

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

In the example \(5(4 + 3)\), begin by solving the expression inside the parentheses, \(4 + 3\), which equals 7.
Next, multiply 5 by the result, which is 7, giving the final value of 35.
Following the order of operations ensures that you solve expressions correctly, preventing errors and avoiding any confusion. Keep practicing this method to become more efficient and confident in solving algebraic expressions!