Problem 58 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \((L e s s o n \quad 1.1)\). $$(8) (a) when a=4$$

Step-by-Step Solution

Answer

\('32'\)

Step 1: Understand and analyze the given expression

The given algebraic expression is 8 * a. In this expression, 'a' is a variable and '8' is a constant.

Step 2: Substitute the given value of a into the expression

We are given, 'a' equals 4. Hence substituting 'a' with '4' in the given expression we get: 8 * 4.

Step 3: Calculate the result

Performing multiplication, we get 8 * 4 = 32

Key Concepts

Substituting Variables

When it comes to evaluating algebraic expressions, one of the crucial steps is substituting variables with their given values. This substitution is the foundation for simplifying an expression and moving closer to the solution.

Take, for example, the simple algebraic expression provided in the exercise, which is initially written as 8 * a. Here, 'a' represents a variable—a placeholder for a number that can vary or change. It's like a blank space in a sentence that you need to fill in to complete the thought. When you are given a specific value for a variable, like 'a' equals 4 in this exercise, you replace every instance of 'a' with the number 4 in the expression.

This transformation from an abstract expression into a more concrete form is pivotal in making the algebraic expression ready for further mathematical operations such as addition, subtraction, multiplication, or division.

Solving Algebraic Expressions

Once variables are substituted, the next step is to solve the algebraic expression. This involves carrying out the prescribed mathematical operations using the standard order of operations, sometimes remembered by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

However, in the given exercise, the operation is straightforward as it only involves multiplication. After substituting the variable 'a' with the number 4, the expression becomes 8 * 4. No other operations are competing for precedence, so you can proceed directly to calculating the product. Solving algebraic expressions can become more complex with the inclusion of more variables and operations, but the fundamental approach remains: substitute, respect the order of operations, and calculate.

Performing Multiplication

Multiplication is one of the primary operations in algebra that we use to evaluate expressions. It entails finding the product of two numbers—when one number is multiplied by another, we want to know how much they collectively amount to.

For instance, in our exercise, after substituting we have 8 * 4. Performing this multiplication involves repeatedly adding the number 8, four times: which gives us 8 + 8 + 8 + 8. The resulting sum, or the product in the case of multiplication, is 32.

The exercise demonstrates a basic but vital concept: multiplication is essentially repeated addition. In a broader context, when dealing with more complex expressions, this operation could be a stepping stone to finding an overall solution and may require additional steps such as distributing over addition or understanding the properties of multiplication, including the commutative, associative, and distributive properties.